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Title: business statistics
Description: The above apploded files are my class notes for last semister. Best in the class of of business and economics students
Description: The above apploded files are my class notes for last semister. Best in the class of of business and economics students
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P
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Box 342-01000 Thika
Email: Info@mku
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ke
Web: www
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ac
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S
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W
Norton and Company
Enns Phillip G, (1985); Business Statistical Methods ad Application; Homewood Richard D Irwin Inc
Module compiled by Charles Karuga
2
Table of content
Course content
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3
CHAPTER 1: INTRODUCTION
...
13
CHAPTER 3: ORGANIZATION AND REPRESENTATION OF DATA
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47
CHAPTER 5: MEASURES OF DISPERSION
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122
CHAPTER 7: RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS
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176
APPENDIX
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Objectives
a)
b)
c)
d)
e)
f)
Define statistics and explain its uses
...
Explain why statistics is distrusted
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Explain the types of variables
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1
...
"
This definition clearly points out four stages in a statistical investigation, namely:
1) Collection of data
2) Presentation of data
3) Analysis of data
4) Interpretation of data
In addition to this, one more stage i
...
organization of data is suggested
Definition:
4
Business statistics is the science of good decision making in the face of uncertainty and is used
in many disciplines such as financial analysis, econometrics, auditing, production and operations
including services improvement, and marketing research
...
2 Uses of Statistics
a) To present the data in a concise and definite form : Statistics helps in classifying and
tabulating raw data for processing and further tabulation for end users
...
, or by condensing the data with the help
of means, dispersion etc
...
d) In forming policies: It helps in forming policies like a production schedule, based on the
relevant sales figures
...
e) Enlarging individual experiences: Complex problems can be well understood by
statistics, as the conclusions drawn by an individual are more definite and precise than
mere statements on facts
...
3 Limitations of Statistics
1
...
Since statistics deals with
aggregates of facts, it cannot be used to study the changes that have taken place in
individual cases
...
But the wages of workers of that industry can be
used statistically
...
But the average
marks or the average height of your class has statistical relevance
...
Statistics cannot be used to study qualitative phenomenon like morality, intelligence,
beauty etc
...
However, it may be possible to analyze such
problems statistically by expressing them numerically
...
3
...
They are true only under certain conditions
...
4
...
Therefore, they can
be used only if mathematical accuracy is not needed
...
Statistics, being dependent on figures, can be manipulated and therefore can be used only
when the authenticity of the figures has been proved beyond doubt
...
3 Distrust of Statistics
It is often said by people that, "statistics can prove anything
...
A Paris banker said, "Statistics is
like a miniskirt, it covers up essentials but gives you the ideas
...
The following reasons account for such views about statistics
...
Figures are convincing and, therefore people easily believe them
...
They can be manipulated in such a manner as to establish foregone conclusions
...
The wrong representation of even correct figures can mislead a reader
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Reading this one
would form the opinion that Jane is decidedly a better worker than John
...
Thus while working with statistics one should not only avoid
outright falsehoods but be alert to detect possible distortion of the truth
...
4 Types of Statistics
Broadly speaking, statistics may be divided into two categories, ie descriptive and inferential
statistics
...
Typically, in most research conducted on groups of people, you will use both descriptive
and inferential statistics to analyze your results and draw conclusions
...
4
...
Descriptive statistics do not, however, allow us to make conclusions beyond the data we
have analyzed or reach conclusions regarding any hypotheses we might have made
...
Descriptive statistics are very important, as if we simply presented our raw data it would be hard
to visualize what the data was showing, especially if there was a lot of it
...
For example, if we had the results of 100 pieces of students'
coursework, we may be interested in the overall performance of those students
...
Descriptive statistics allow us to do this
...
In this case, the frequency distribution is
simply the distribution and pattern of marks scored by the 100 students from the lowest to
the highest
...
7
Measures of spread: these are ways of summarizing a group of data by describing how
spread out the scores are
...
However, not all students will have scored 65 marks
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Some will be lower and others higher
...
To describe this spread, a number of
statistics are available to us, including the range, quartiles, absolute deviation, variance
and standard deviation
...
e
...
e
...
e
...
1
...
2 Inferential Statistics
Whilst descriptive statistics examine our immediate group of data (for example, the 100 students'
marks), inferential statistics aim to make inferences from this data in order to make conclusions
that go beyond this data
...
For example, a Board of Examiners may want to compare the performance of 1000 students that
completed an examination
...
The 1000
students represent our "population"
...
Instead, we can choose to
examine a "sample" of these students and then use the results to make generalizations about the
performance of all 1000 students
...
Since we are looking to compare boys and girls, we may randomly select 100
girls and 100 boys in our sample
...
8
1
...
Bias: - Bias means prejudice or preference of the investigator, which creeps in
consciously and unconsciously in proving a particular point
...
Generalization:- Some times on the basis of little data available one could jump to a
conclusion, which leads to erroneous results
...
Wrong conclusion:- The characteristics of a group if attached to an individual member of
that group, may lead us to draw absurd conclusions
...
Incomplete classification:- If we fail to give a complete classification, the influence of
various factors may not be properly understood
...
There may be a wrong use of percentages
...
Technical mistakes may also occur
...
An inconsistency in definition can even exist
...
Wrong causal inferences may sometimes be drawn
...
6 Types of Variables
1
...
1 Discrete Variable
A discrete variable is one that cannot take on all values within the limits of the variable
...
The
variable cannot have the value 1
...
A variable such as a person's height can take on any value
...
Statistics computed from discrete variables have many more possible values than the discrete
variables themselves
...
117 even though 3
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1
...
2Continuous Variable
A continuous variable is one for which, within the limits the variable ranges, any value is
possible
...
13 minutes etc
...
The variable "Number of correct
9
answers on a 100 point multiple-choice test" is not a continuous variable since it is not possible
to get 54
...
A variable that is not continuous is called "discrete"
1
...
7
...
No quantitative
information is conveyed and no ordering of the items is implied
...
Religious preference, race, and sex are all examples of
nominal scales
...
The main statistic computed is the mode
...
1
...
2 Ordinal Scale
Measurements with ordinal scales are ordered in the sense that higher numbers represent higher
values
...
For example, on a
five-point rating scale measuring attitudes toward gun control, the difference between a rating of
2 and a rating of 3 may not represent the same difference as the difference between a rating of 4
and a rating of 5
...
The lowest point on the rating scale in the example was arbitrarily chosen to be 1
...
1
...
3 Interval Scale
On interval measurement scales, one unit on the scale represents the same magnitude on the trait
or characteristic being measured across the whole range of the scale
...
Interval scales do not have a "true" zero point, however, and therefore it is not
possible to make statements about how many times higher one score is than another
...
True interval measurement is somewhere between rare and
nonexistent in the behavioral sciences
...
A good example of an interval scale is the Fahrenheit
scale for temperature
...
1
...
4 Ratio Scale
Ratio scales are like interval scales except they have true zero points
...
This scale has an absolute zero
...
Chapter Review Questions
1
...
Define Business statistics
3
...
4
...
5
...
6
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The mass of a bull
ii
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The rank of an army officer
iv
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11
Reference
i
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Pg13-14, Pg 71-93
ii
...
A Saleemi
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Pg 237244
iii
...
Objectives
a) Distinguish between primary and secondary data
...
c) Define sampling and explain various methods of sampling
For any statistical enquiry, the basic objective is to collect facts and figures relating to a
particular phenomenon for further statistical analysis
...
2
...
You
have to either trust the honesty of the people surveyed or build in self-verifying questions
(e
...
questions 9 and 24 ask basically the same thing but using different words - different
answers may indicate the surveyed person is being inconsistent, dishonest or inattentive)
...
They also show non-verbal
communication such as face-pulling, fidgeting, shrugging, hand gestures, sarcastic
expressions that add further meaning to spoken words
...
g
...
logs (e
...
fault logs, error logs, complaint logs, transaction logs)
...
Can yield lots of valuable data
about system performance over time under different conditions
...
It's
like cooking something yourself
...
Secondary data is collected from external sources such as:
TV, radio, internet
magazines, newspapers
reviews
research articles
stories told by people you know
There's a lot more secondary data than primary data, and secondary data is a whole lot cheaper
and easier to acquire than primary data
...
Who collected it? Can they be trusted? Did they do any
preprocessing of the data? Is it biased? How old is it? Where was it collected? Can the data be
verified, or does it have to be taken on faith?
Often secondary data has been pre-processed to give totals or averages and the original details
are lost so you can't verify it by replicating the methods used by the original data collectors
...
Secondary data is
cheap and easy to collect, but must be treated with caution
...
2 Methods of collecting Primary data
14
In primary data collection, you collect the data yourself using methods such as interviews and
questionnaires
...
There are many methods of collecting primary data and the main methods include:
questionnaires
interviews
focus group interviews
observation
case-studies
diaries
critical incidents
Portfolios
...
We briefly outline these methods but you should also read around
the various methods
...
2
...
Advantages:
Can be used as a method in its own right or as a basis for interviewing or a telephone
survey
...
Can cover a large number of people or organisations
...
Relatively cheap
...
Avoids embarrassment on the part of the respondent
...
Possible anonymity of respondent
...
Disadvantages:
Design problems
...
Historically low response rate (although inducements may help)
...
Require a return deadline
...
Assumes no literacy problems
...
Not possible to give assistance if required
...
Replies not spontaneous and independent of each other
...
For example, perhaps because it is too long, too complex, uninteresting, or too
personal
2
...
2 Interviews
Interviewing is a technique that is primarily used to gain an understanding of the underlying
reasons and motivations for people’s attitudes, preferences or behaviour
...
They can be conducted at work, at
home, in the street or in a shopping centre, or some other agreed location
...
Good response rate
...
Possible in-depth questions
...
Can investigate motives and feelings
...
Characteristics of respondent assessed – tone of voice, facial expression, hesitation, etc
...
17
If one interviewer used, uniformity of approach
...
Disadvantages:
Need to set up interviews
...
Geographic limitations
...
Normally need a set of questions
...
Embarrassment possible if personal questions
...
If many interviewers, training required
...
2
...
Case-studies involve measuring what is
there and how it got there
...
It can enable the researcher to explore,
unravel and understand problems, issues and relationships
...
The case looked at may be unique and, therefore
not representative of other instances
...
The case-study
18
approach is often done to make practical improvements
...
The case-study method has four steps:
1
...
2
...
3
...
The background information collected will have been analysed for
possible hypotheses
...
This step aims to eliminate possibilities which conflict with the evidence
collected and to gain confidence for the important hypotheses
...
4
...
The aim is to check that the hypotheses tested actually work out in
practice
...
The case-study enables rich information to be gathered from which potentially useful hypotheses
can be generated
...
It is also inefficient in researching
situations which are already well structured and where the important variables have been
identified
...
2
...
4 Diaries
A diary is a way of gathering information about the way individuals spend their time on
professional activities
...
Advantages:
19
Useful for collecting information from employees
...
Allows the researcher freedom to move from one organization to another
...
Diaries can be used as a preliminary or basis for intensive interviewing
...
Disadvantages:
Subjects need to be clear about what they are being asked to do, why and what you plan
to do with the data
...
Some structure is necessary to give the diarist focus, for example, a list of headings
...
Progress needs checking from time-to-time
...
Analyses problems, so you need to consider how responses will be coded before the
subjects start filling in diaries
...
2
...
The compilation of problem portfolios is
20
recording information about how each problem arose, methods used to solve it, difficulties
encountered, etc
...
What
proportion of time is occupied in checking; in handling problems given by others; on selfgenerated problems; on ‘top-priority’ problems; on minor issues, etc? The main problem with
this method and the use of diaries is getting people to agree to record everything in sufficient
detail for you to analyze
...
3 Sampling
Collecting data is time consuming and expensive, even for relatively small amounts of data
...
Because of the time
and cost elements the amount of data you collect will be limited and the number of people or
organizations you contact will be small in number
...
Sampling theory says a correctly taken sample of an appropriate size will yield results that can be
applied to the population as a whole
...
How is a sample taken correctly?
2
...
It is like
answering the question ‘How long is a piece of string’? It all depends on the circumstances
...
You therefore need to know some of the basics of sampling
...
The theory of sampling is based on random samples – where all items in the population have the
same chance of being selected as sample units
...
This
information is usually in the form of an alphabetical list – called the sampling frame
...
2
...
1 Simple random sampling
Simple random sampling can be carried out in two ways – the lottery method and using random
numbers
...
This is basically similar to a game of bingo or the national lottery
...
Alternatively random numbers can be used
...
An example
of these is:
03 47 43 73 86
36 96 47 36 61
97 74 24 67 62
42 81 14 57 20
16 76 62 27 66
56 50 26 71 07
12 56 85 99 26
96 96 68 27 31
22
55 59 56 35 64
38 04 80 46 22
Random numbers tend to be written in pairs and blocks of 5 by 5 to make reading easy
...
The numbers can be read in any direction but
they should be read as a single string of digits i
...
left to right as 0, 3, 4, 7 etc’, or top to bottom
as 0, 9, 1, 1, 5, 3, 7, … etc
...
The random number method involves:
Allocating a number to each person on the list (each number must consist of the same
number of digits so that the tables can be read consistently)
...
Read off the digits
...
For the example of selecting nine people at random from 90:
a) The sampling frame is the list of 90 people
...
Note that each number has
two digits and the numbering starts from 00
...
Then the person that has been numbered 16 is the first
sample unit
...
23
This procedure is repeated until the nine people have
been identified
...
Simple random number sampling is used as the basis for many other sampling methods, but has
two disadvantages:
A sampling frame is required
...
The procedure is unbiased but the sample may be biased
...
2
...
2 Stratified Sampling
To overcome the second problem above, a stratified sample can be taken
...
For example, suppose the 90 people consist of 30 men and 60 women
...
e
...
e
...
The three men and six women would then be selected by simple random sampling e
...
, random
numbers
...
g
...
Also a
more detailed sampling frame is required
...
3
...
For example, for the population 90
and sample of nine:
Split the sampling frame in to nine equal groups
...
e
...
Suppose this number is 6
...
Then the 16th, 26th, 36th, 46th, 56th, 66th, 76th, and 86th people are the remaining sample units
...
Systematic sampling can be used by selecting a random number say 25
...
The 50th person to enter is the second sample unit
...
This approach usually generates a good cross section of the population
...
Review Questions
1
...
2
...
Indicate the situations in which each
of these methods should be used
...
What is sampling?
4
...
Discuss the various sampling methods
...
Pg 71-93
b) Business Calculations and statistics simplified by N
...
Revised Edition
...
Objectives
a)
Explain the general principle of constructing diagrams
...
Bar charts
ii
...
Pie charts
iv
...
Ogives
vi
...
Box and whisker plots
3
...
Forecasting also becomes easier with the help of
graph
...
3
...
The diagrams should be simple
...
Each diagram must be given a clear, concise and suitable title without damaging clarity
...
A proper proportion between height and width must be maintained in order to avoid an
unpleasant look
...
Select a proper scale; it should be in even numbers or in multiples of five or ten
...
g
...
But no fixed rule
...
In order to clear certain points, always put footnotes
...
An index, explaining different lines, shades and colors should be given
...
Diagrams should be absolutely neat and clean
...
Above all the chart must be honest
...
W
...
3
...
3
...
For example sales, production, population figures etc
...
Since these are of the same width and vary only in
heights ( or lengths ), it becomes very easy for readers to study the relationship
...
A bar chart can be either vertical or horizontal; vertical
bars are more popular
...
Country
Birth rate
Country
Birth rate
India
33
China
40
Germany
15
New Zealand
30
U
...
20
Sweden
15
28
Represent the above data by a suitable diagram
...
Such diagrams are also known as component
bar diagrams
...
3
...
A common and helpful arrangement is that of presenting each bar in the order of
magnitude with the largest component at the bottom and the smallest at the top
...
Illustration:- During 1968 - 71, the number of students in University ' X ' are as follows
...
Year
Arts
Science
Law
Total
1968-69 20,000
10,000
5,000 35,000
1969-70 26,000
9,000
7,000 42,000
1970-71 31,000
9,500
7,500 48,000
29
3
...
3 Multiple Bar Diagram
This method can be used for data which is made up of two or more components
...
The height of each bar represents the
actual value of the component
...
Where
changes in actual values of component figures only are required, multiple bar charts are used
...
Year
Export
Import
1927 - 28
319
250
1928 - 29
339
263
1929 - 30
345
258
1930 - 31
308
206
30
Represent the data by a suitable diagram
3
...
4 Deviation Bar Charts
Deviation bars are used to represent net quantities - excess or deficit i
...
net profit, net loss, net
exports or imports, swings in voting etc
...
Positive values lie above the base line and negative values lie below it
...
3
...
It is therefore sufficient to draw angles at the center, proportional to the
original figures
...
For example, let the total be 1000 and one of the component be 200, then the angle will be
In general, angle of sector at the center corresponding to a component
ii) When a statistical phenomenon is composed of different components which are numerous
(say four or more components), bar charts are not suitable to represent them because, under this
situation, they become very complex and their visual impressions are questioned
...
It is a circular diagram which is a circle (pie) divided by the radii,
into sectors ( like slices of a cake or pie )
...
They are often used to
present financial information
...
g
...
e
...
A pie chart is a circular chart in which the circle is divided into sectors
...
Example 9
A family's weekly expenditure on its house mortgage, food and fuel is as follows:
Draw a pie chart to display the information
...
Percentage of weekly expenditure on:
33
To draw a pie chart, divide the circle into 100 percentage parts
...
It is simple to read a pie chart
...
For example, the weekly expenditure of the family on
food is 37
...
A pie chart is used to compare the different parts that make up a whole amount
...
5 Graphs
A graph is a visual representation of data by a continuous curve on a squared ( graph ) paper
...
Graphs of Frequency Distributions:The methods used to represent a grouped data are :1
...
Frequency Polygon
3
...
Ogive or Cumulative Frequency Curve
34
3
...
1
Histogram
It is defined as a pictorial representation of a grouped frequency distribution by means of
adjacent rectangles, whose areas are proportional to the frequencies
...
The rectangles are constructed such that the height of
each rectangle is proportional to the frequency of the class and width is equal to the length of the
class
...
In case
of classes having unequal widths, rectangles too stand on unequal widths (bases)
...
As the rectangles are
adjacent leaving no gaps, the class-intervals become of the inclusive type, adjustment is
necessary for end points only
...
Let us say you sold total 31 books
at this book-fair at the following prices
...
2, $ 1, $ 2, $ 2, $ 3, $ 5, $ 6, $ 17, $ 17, $ 7, $ 15, $ 7, $ 7, $ 18, $ 8, $ 10, $ 10, $ 9, $ 13, $
11, $ 12, $ 12, $ 12, $ 14, $ 16, $ 18, $ 20, $ 24, $ 21, $ 22, $ 25
...
Divide this range into number of groups, class intervals
...
Our first class-interval includes the lowest price of the data and, the last-interval of course
includes, the highest price
...
For example you have class intervals as 0-5, 5-10, 10-15 and so on, then
the price $10 falls in both 5-10 and 10-15
...
Therefore now we have distribution of books at a book-faire
35
Class-interval
Frequency
$ 1- $ 5
6
$6 - $10
8
$11 - $15
10
$16 - $20
3
$21 - $25
4
Total
n = sum fi = 31
Note that each class-interval is of equal width i
...
$5 inclusive
...
36
LECTURE 5
Methods of Presenting Data (continuation from lecture 4)
3
...
2
Frequency Distribution (Curve):Frequency distribution curves are like frequency polygons
...
The
frequency curve for the above data is shown as:
3
...
3
Ogives or Cumulative Frequency Curves
When frequencies are added, they are called cumulative frequencies
...
To construct an Ogive:1) Add up the progressive totals of frequencies, class by
class, to get the cumulative frequencies
...
37
3) Join the points by a smooth curve
...
In most of the cases it looks like 'S'
...
(A) Less than Ogive:- To plot a less than ogive, the data is arranged in ascending order of
magnitude and the frequencies are cumulated starting from the top
...
(B) Greater than Ogive:- To plot this ogive, the data are arranged in the ascending order of
magnitude and frequencies are cumulated from the bottom
...
Illustrations:- On a graph paper, draw the two ogives for the data given below of the I
...
of 160
students
...
of students:
2
7
12
28
42
110 - 120 120 - 130 130 - 140 140 - 150 150 - 160
36
18
10
4
38
1
39
Uses: - Certain values like median, quartiles, deciles, quartile deviation, coefficient of skewness
etc
...
It can be used to find the percentage of items having values less
than
...
6 Stem and Leaf Diagram
A stem and leaf diagram provides a visual summary of your data
...
There are three steps for drawing a tem and leaf diagram
...
Split the data into two pieces, stem and leaf
...
Arrange the stems from low to high
...
Attach each leaf to the appropriate stem
...
Note that with the data written in this way
you can see what the modal class is (the one with the most values
...
To change this into a stem and leaf diagram, we just simplify it a little
...
) we write '14' and call this the 'stem' and then write
3, 3, 4, 3,
...
We would usually, however, write the leaves in order
(with the smallest first)
...
40
So we finish up with:
3
...
1 Back-to-back stem and leaf diagram
Back-to-back stem plots are used to compare two distributions side-by-side
...
The center column contains
the stems
...
The
numbers for the leaves of the distribution in the leftmost column are aligned to the right and are
listed in increasing order from right to left
...
BOYS
GIRLS
3
4
40
5
4
1
2
8
5
4
3
5
5
0
50
2
3
5
8
9
4
5
2
2
3
3
4
5
60
3
5
6
5
5
2
8
0
2
70
0
3
3
3
1
3
4
80
3
6
4
4
4
9
90
3
4
41
KEY:
40
5 =45
Can you comment on the shape of the distribution of the two sets of data?
3
...
It displays a number of statistics like, median, lower
quartile (Q1), upper quartile (Q3), Inter-quartile range (IQR)
...
Example
Statistics CAT scores of 12 students are as follows:10, 22, 24, 27, 31, 33, 39, 40, 42, 43, 44, 45
Draw a box and whisker plot to represent the above scores
...
10, 22, 24, 27, 31, 33, 39, 40, 42, 43,
44, 45
1) Since n = 12 (total items)
the two middle scores are
12
12 2
6 th and
7 th
2
2
i
...
33 and 39 respectively
...
e
...
i
...
25th percentile
...
i
...
75th percentile
...
ii) The left side of this box indicates the lower quartile (Q1)
...
iv) A straight line is then drawn from the lowest value of this
distribution through the box to the highest value of this
distribution
...
Then the above CAT score in box-plot will look like this:
0 10 20 30 40 50 60
Chapter Review Questions
43
1
...
10
8
6
Number of
families 4
2
0
(a)
3
4
5
6
7
8
Number of people in a family
9
10
How many families are represented?
(b) Write down the mode of the distribution
...
2
...
The results are given in the table below
...
(b) Draw a cumulative frequency curve
...
3
...
Ages
Number of
smokers
20 ≤ x < 30
5
30 ≤ x < 40
4
40 ≤ x < 50
3
50 ≤ x < 60
2
60 ≤ x < 70
3
(a) Calculate an estimate of the mean smoking age
...
4
...
180 184 195 177 175 173 169 167 197 173 166 183 161 195 177
192 161 165
45
Represent the data by a stem and leaf diagram
...
The following stem and leaf diagram gives the heights in cm of 39 schoolchildren
...
13
2, 3, 3, 5, 8,
14
1, 1, 1, 4, 5, 5, 9,
15
3, 4, 4, 6, 6, 7, 7, 7, 8, 9, 9,
16
1, 2, 2, 5, 6, 6, 7, 8, 8,
17
4, 4, 4, 5, 6, 6,
18
0,
(a) (i)
State the lower quartile height,
(ii)
State the median height
(iii)
State the upper quartile height
...
References
Research methods by Mugenda Olive M and Mugenda Abel G
...
A Saleemi
...
Pg 275-285
Essentials of statistics for Business and Economics by Anderson Sweety Williams Pg 22-34
46
LECTURE4
CHAPTER 4: ANALYSIS AND INTERPRETATION OF DATA
Purpose
To examine various measures of central tendency
...
ii
...
iv
...
vi
...
1 Introduction
In the previous chapter, we have studied how to collect raw data, its classification and tabulation
in a useful form, which contributes in solving many problems of statistical concern
...
We may reduce the entire distribution to
one number which represents the distribution
...
Since such typical values tend to lie centrally within a set of
observations when arranged according to magnitudes, averages are called measures of central
tendency
...
This is of great importance, both theoretically and practically
...
A
...
Bowley correctly stated, "Statistics may rightly be called the science of averages
...
For example, we may say that
Okanga is an average boy of my class; we may talk of an average American, average income,
etc
...
However, in statistics the term average has a different
meaning
...
We therefore, consider the Arithmetic mean
...
2 Arithmetic Mean
This is the most commonly used average which you have also studied and used in lower grades
...
Arithmetic mean is the amount secured by dividing the sum of values of the items in a series by
their number
...
Thus, you should add all observations (values of all items) together and divide this sum by the
number of observations (or items)
...
2
...
, xn then the Arithmetic mean is
obviously
We shall use the symbol x (pronounced as x bar) to denote the Arithmetic mean
...
The symbol xi will be used to denote, in general the 'i' th
n
observation
...
+ xn will be represented by
x or x simply
i 1
i
i
Therefore the Arithmetic mean of the set x1 + x2 + x3 +
...
Example A variable takes the values as given below
...
Solution: Arithmetic mean =
x
i
n
49
x = 110 + 117 + 129 +195 + 95 +100 +100 +175 +250 + 750 = 2021
i
and n = 10
Indirect Method (Assumed Mean Method)
u
u
i
n
Where u xi A
and
A = Assumed Mean
Mean= x A u
Calculations:
Let A = 175 then
ui = -65, -58, -46, +20, -80, -75,-75, +0, + 75, +575
= 670 - 399
= 271/10 = 27
...
1
= 202
...
Sonko’s earnings for the past week were:
Monday
$ 450
Tuesday
$ 375
Wednesday
$ 500
Thursday
$ 350
Friday
$ 270
50
Find his average earning per day
...
Sonko’s average earning per day is $389
...
For this, first you assume a mean (called as the assumed mean)
...
Now
find the deviations of all the values of x from A
...
Calculate the Arithmetic mean
...
(as
...
2
...
In the discrete series, every term (i
...
value of x) is multiplied by its
corresponding frequency f i xi and then their total (sum) is found f i xi
...
i i
52
Therefore, if the observations x1 + x2 + x3 +
...
+ fn times,
then we have:
Arithmetic mean can be found by
The formulae for Arithmetic mean by direct method and by the short-cut methods are as follows:
Direct method
Short-cut method
and u = xi – A
Therefore,
Example Find the mean of the following 50 observations
...
The arithmetic mean is
Example Eight coins were tossed together and the number of times they fell on the side of heads
was observed
...
Calculate then
mean by:
i) Direct method ii) Short-cut method
x:
0
1
2
f:
1
9
26 59
3
4
5
6
7
8
72
52
29
7
1
54
Solution:
Example Find the arithmetic mean for the following :
Marks below : 10 20 30 40 50 60 70 80
No
...
Marks
Mid-values
Frequencies
U = X -A
fiui
xi
c
...
f
...
This method is employed in place of the
Short-cut method
...
All the class marks happen to be multiples of c, since all class
intervals are equal
...
Theorem If x1, x2 , x3,
...
fn
respectively and if each xi is expressed in terms of the new variable ui by the relation
56
xi = A + c ui then, with the usual notation, we have
where
and
This method is also known as the "Coding method
...
of employees : 8 23 51 81 103 113 117 120
Solution :
57
Example From the following data, of the calculation of arithmetic mean, find the missing item
...
of
workers
25
17
13
15
14
8
7
Mean wage $ 115
...
2
...
The sum of the deviations, of all the values of x, from their arithmetic mean, is zero
...
The product of the arithmetic mean and the number of items gives the total of all items
...
If x1 and x2 are the arithmetic mean of two samples of sizes n1 and n2 respectively then,
the arithmetic mean x of the distribution combining the two can be calculated as
This formula can be extended for still more groups or samples
...
Find the average marks of all the 150 students, taken together
...
Batch - I Batch - II Batch - III
A
...
of students n1 = 70
= 55
n2 = 50
= 45
n3
= 30
Example The mean of a certain number of observations is 40
...
Find the number of items in the original
data
...
total of n values
...
New
42n + 84 = 40n + 114
2n = 30
n = 15
Therefore, the number of items in the original data = 15
...
Find the number of observations
and their mean
...
Note
4 4n and x
i
7 3
62
x 4n 72
i
therefore,
x
i
7 3
Subtracting the two equations we get,
(-)
(+)
(+)
3n=75
n = 25
Putting n = 25 in
Now Mean is given by x
, we get
x
i
n
172
688
25
Example The mean weight of 98 students is found to be 50 kg
...
Calculate the
correct mean
...
\The correct
Also the correct
Therefore, the correct mean
x
correct f i xi
correct f i
4970
100
49
...
Show that the arithmetic mean is a
64
S
...
2
...
It is rigidly defined
...
2
...
Hence it is very popular
...
It is based on all the observations; so that it becomes a good representative
...
It can be easily used for comparison
...
It is capable of further algebraic treatment such as finding the sum of the values of the
observations, if the mean and the total number of the observations are given; finding the
combined arithmetic mean when different groups are given etc
...
It is not affected much by sampling fluctuations
...
2
...
It is affected by outliers or extreme values
...
) mean of 10, 15,
25 and 500 is x
10 15 25 500
137
...
67(approx)
3
Due to the outlier 500 the A
...
5
...
mean is not a good representative of the given data
...
It is a value which may not be present in the given data
...
Many a times it gives absurd results like 4
...
4
...
5
...
66
4
...
Thus, it is the value of the middle item and divides the series in to
equal parts
...
" For example, the daily wages of 7 workers are 5, 7, 9, 11, 12, 14 and 15 dollars
...
The fourth term i
...
$11 is the median
...
3
...
Set the individual series either in the ascending (increasing) or in the descending
(decreasing) order, of the size of its items or observations
...
If the total number of observations be 'n' then
A
...
If 'n' is even, the median
=
Example The following figures represent the number of books issued at the counter of a
Statistics library on 11 different days
...
Calculate the median
...
Now the total number of items 'n'= 11 (odd)
Therefore, the median = size of
item
=
size of
=
size of 5th item
item
= 98 books per day
Example The population (in thousands) of 36 metropolitan cities are as follows :
2468, 591, 437, 20, 213, 143, 1490, 407, 284, 176, 263, 19, 181, 777, 387, 302, 213, 204, 153,
733, 391, 176 178, 122, 532, 360, 65, 260, 193, 92, 672, 258, 239, 160, 147, 151
...
Solution:
Arranging the terms in the ascending order as:
20, 65, 92, 131, 142, 143, 147, 151, 153, 160, 169, 176, 178, 181, 193, 204, (213, 39), 258, 263,
260, 384, 302, 360, 387, 391, 407, 437, 522, 591, 672, 733, 777, 1490, 2488
...
the median
=
68
4
...
2 Median In Discrete Series
Steps :
1
...
2
...
3
...
If 'n' =
(odd) then,
Median = size of
B
...
Size
: 8 10 12 14 16 18 20
Frequency : 7
7
12
28
10 9 6
69
Solution:
Therefore, the median =
=
= size of 38th item
In the order of the cumulative frequency, the 38th term is present in the 50th cumulative
frequency, whose size is 14
...
3
...
Determine the particular class in which the value of the median lies
...
After ascertaining the class in which median lies, the following formula is used for
determining the exact value of the median
...
= upper limit of the median class
c
...
Example Calculate the median for the following and verify it graphically
...
of person
:
70
80
180
150
20
71
Solution:
Median =
Here
= 30,
= 35,
= 250, c
...
= 150 and f = 180
Therefore, Median
72
Sometimes the series is given in the descending order of magnitude
...
of students : 10
12
40
30
8
73
Solution :
By interpolation
74
Arranging the series in the descending order (as it is given)
th
n
Median = size of item = size of 50th item which lies in (20 -30) class-interval
...
If the original series is in inclusive type, first convert it into the exclusive type and then
find its median
...
What is the median?
75
Minutes/Weeks: 0-99 100-199 200-299 300-399 400 - 499 500 - 599 600 & more
No
...
3
...
It is rigidly defined
...
It is easy to calculate and understand
...
It is not affected by extreme values like the arithmetic mean
...
The median would be $2600 while
the arithmetic mean would be $3020
...
It can be found by mere inspection
...
It is fully representative and can be computed easily
...
It can be used for qualitative studies
...
Even if the extreme values are unknown, median can be calculated if one knows the
number of items
...
It can be obtained graphically
...
3
...
It may not be representative if the distribution is irregular and abnormal
...
It is not capable of further algebraic treatment
...
It is not based on all observations
...
It is affected by sample fluctuations
...
The arrangement of the data in the order of magnitude is absolutely necessary
...
4 Mode
It is the size of that item which possesses the maximum frequency
...
It is the most common value
...
77
4
...
1 Ungrouped Data
Individual series : The mode of this series can be obtained by mere inspection
...
Example Locate mode in the data 7, 12, 8, 5, 9, 6, 10, 9, 4, 9, 9
Solution : On inspection, it is observed that the number 9 has maximum frequency
...
Note that if in any series, two or more numbers have the maximum frequency, then the mode
will be difficult to calculate
...
4
...
2 Grouped Data
Steps :
1
...
2
...
Daily wages in $ : 20 -25 25-30 30-35 35-40 40-45 45-50
78
No
...
Solution:
Here the maximum frequency is 12, corresponding to the class interval (35 - 40) which is the
modal class
...
22
79
4
...
3 Merits of Mode
1
...
2
...
3
...
Everyone is used to the idea of average size of a garment, an
average American etc
...
It is not isolated like the median as it is the most common item
...
Like the Average mean, it is not a value which cannot be found in the series
...
It is not necessary to know all the items
...
7
...
4
...
4 Demerits of Mode
1
...
2
...
3
...
4
...
5
...
4
...
e
...
Conversely, when values of mean, median and
mode are not equal the distribution is known as asymmetrical or skewed distribution
...
In such distributions the distance between the mean and
median is about one-third of the distance between the mean and mode, as will be clear from the
diagrams 1 and 2
...
Example
Given median = 20
...
Solution:
4
...
Each data value (Xi) has a weight assigned to it (Wi)
...
The formula is
There are several reasons why you might want to use a weighted mean
...
Each individual data value might actually represent a value that is used by
multiple people in your sample
...
2
...
To restore balance, you would place less weight on
the over represented segments of the population and greater weight on the
underrepresented segments of the population
...
Some values in your data sample might be known to be more variable (less
precise) than other values
...
Example
Joan gets quiz grades of 79, 82, and 69
...
Find the weighted mean
if the quizzes each count for 10% and the final exam counts for 70% of the final grade
...
5%
XW
4
...
A common example when the geometric mean is use is when averaging growth rates
...
M n [ x1 ][ x2 ][ x3 ]
...
M 3 [3][ 25][45]
= 15
The geometric mean cannot be calculated if we have negative or zero observations
...
Take the arithmetic mean of the following salaries: - in thousands of shillings per month
6, 8, 10, 10,10,12,16
Arithmetic mean = Kshs 10, 286 per month to the nearest shilling
...
M 7 [6][8][10][10][10][12][16]
7 9216000
= 9
...
9884 per month
(To the nearest shilling)
Given the following salaries (i
...
in thousands of Kshs) in accompany per annum (p
...
The geometric mean is:G
...
564
11
...
11
...
The geometric
mean is useful when only a few items in a distribution are changing: it’s in the circumstances
more stable than the arithmetic mean
...
e
...
300,000 + 400,000 = 700,000
2
2
= 350,00
Here, we are making an assumption the population grows by the same number each year which
is not correct
...
The
geometric mean for 1985 would be:= 2√ 300,000 x 400,000
= 371,080
4
...
Like arithmetic mean and geometric mean, harmonic
84
mean is also useful for quantitative data
...
Harmonic mean in mathematical terms is defined as follows:
For Ungrouped Data
H
...
5, 14
...
8, 15
...
1
Solution:
The harmonic mean is calculated as below:
x
1
x
13
...
0758
14
...
0704
14
...
0676
15
...
0658
16
...
0621
Total
x 0
...
M X
H
...
63
0
...
Calculate the Harmonic Mean
...
While the number of students Represent frequencies
...
1538
14
5
0
...
8667
16
7
0
...
1765
Total
f
30
Now we will find the Harmonic Mean as
86
1
x 1
...
9916 15
...
5
2
0
...
5
3
0
...
5
11
0
...
5
20
0
...
5
32
0
...
5
25
0
...
5
7
0
...
60
f 1
...
4368
4
...
It should be rigidly defined
2
...
It should be based on all observations
4
...
5
...
It should be least affected by fluctuations in sampling
...
The mean of the ten numbers listed below is 5
...
4, 3, a, 8, 7, 3, 9, 5, 8, 3
(a)
Find the value of a
...
2
...
2, b, 3, a, 6, 9, 10, 12
Find each of the following
(a)
the value of a;
(b)
the value of b
...
For the set of {8, 4, 2, 10, 2, 5, 9, 12, 2, 6}
(a)
calculate the mean;
(b)
find the mode;
(c)
find the median
...
David looked at a passage from a book
...
Class interval
Frequency
(number of words)
f
1–5
16
6–10
28
11–15
26
16–20
14
21–25
10
26–30
3
31–35
1
36–40
0
41–45
2
(a)
Find the class interval in which the median lies
...
5
...
The results are summarized in the frequency distribution table below
...
The weight in kilograms of 12 students in a class are as follows
...
(b)
Calculate the mean weight;
When one student leaves the class, the mean weight of the remaining 11 students
becomes 70 kg
...
...
An atlas gives the following information about the approximate population of
some cities in the year 2000
...
City
Population in Millions
Melbourne
3
...
2
Nairobi
Paris
9
...
7
Tokyo
28
...
1
The atlas tells us that the mean population for this group of cities is 10
...
(a)
Calculate the population of Nairobi
...
The number of hours that a professional footballer trains each day in the month
of June is represented in the following histogram
...
(b)
Calculate the mean number of hours he trains each day
...
The numbers of games played in each set of a tennis tournament were
9, 7, 8, 11, 9, 6, 10, 8, 12, 6, 8, 13, 7, 9, 10, 9, 10, 11,
12, 8, 7, 13, 10, 7, 7
...
games
frequency
6
2
7
5
8
n
9
4
10
4
11
2
12
2
13
2
(a)
Write down the value of n
...
(c)
What percentage of the sets had more than 10 games?
(d)
What is the modal number of games?
References
Business Calculations and statistics simplified by N
...
Revised Edition
...
Objectives
a) Define a measure of dispersion and differentiate it from a measure of central
tendency
...
c) Define skewness and kurtosis and compute skewness
...
e) Apply these measures in summarizing a business environment
...
1 Introduction
The measures of central tendencies (i
...
means) indicate the general magnitude of the data and
locate only the center of a distribution of measures
...
i) According to Nciswanger, "Two distributions of statistical data may be symmetrical and have
common means, medians and modes and identical frequencies in the modal class
...
"
ii) Simpson and Kafka said, "An average alone does not tell the full story
...
Scatter
around it
...
"
93
From this discussion we now focus our attention on the scatter or variability which is known as
dispersion
...
Students
Group
Group
Group
X
Y
Z
1
50
45
30
2
50
50
45
3
50
55
75
50
50
50
mean
Thus, the three groups have same mean i
...
50
...
Now if one would say that the students from the three groups are of equal capabilities, it is
totally a wrong conclusion then
...
It is thus clear that the measures of the central tendency is alone not
sufficient to describe the data
...
Thus the ’dispersion’ is also known
as the "average of the second degree
...
Griffin and Dr
...
In measuring dispersion, it is imperative to know the amount of variation (absolute measure) and
the degree of variation (relative measure)
...
In the latter case we consider the coefficient of range, the
coefficient mean deviation, the coefficient of variation etc
...
2 Methods of Computing Dispersion
94
Note that, we are going to study some of these and not all
...
2
...
Thus Range (R) = L - S
Coefficient of Range : The relative measure of the range
...
Solution: R = L - S = 790 - 100 = 690
Co-efficient of Range =
5
...
2 Mean Deviation
The mean deviation of a statistical data is defined as the arithmetic mean of the numerical
values of the deviations of items from some average value
...
The mean deviation is generally denoted by M
...
95
Formula for mean deviation for ungrouped data or an individual series is given by
For a frequency distribution, the formula for mean deviation is given by
where M
...
M is the Arithmetic Mean
...
96
Solution
Example (Continuous series) calculate the mean deviation and the coefficient of mean deviation
from the following data using the mean
...
Diff
...
of
years:
students:
0–5
449
97
5 – 10
705
10 – 15
507
15 – 20
281
20 – 25
109
25 – 30
52
30 – 35
16
35 – 40
4
98
Calculation:
1) x
fx
i i
n
22217
...
5(approx)
2123
2) M
...
3) Co efficient of M
...
5
...
3 Variance
The term variance was used to describe the square of the standard deviation
...
Variance is defined as follows:
x x
var
2
i
n
99
5
...
4 Standard Deviation (s
...
)
It is the square root of the arithmetic mean of the square deviations of various values from their
arithmetic mean
...
d (when dealing with a sample) or (when dealing with the
population)
x
Thus s
...
d
where n =
n
f x
i
i
x
n
f
x
i
2
for ungrouped data
2
for grouped data
i
Merits :
(1) It is rigidly defined and based on all observations
...
(3) It is not affected by sampling fluctuations
...
(2) It gives greater weight to extreme values
...
3 Co‐efficient Of Variation ( C
...
)
To compare the variations (dispersion) of two different series, relative measures of standard
deviation must be calculated
...
d
...
V
...
d
...
Remark: It is given as a percentage and is used to compare the consistency or variability of two
more series
...
V
...
V
...
Example Calculate the standard deviation and its co-efficient from the following data
...
xi
(xi - x)
( xi - x )2
A
10
-5
25
B
12
-3
9
C
16
+1
1
D
8
-7
49
E
25
+10
100
101
F
30
+15
225
G
14
-1
1
H
11
-5
16
I
13
-2
4
J
11
-4
16
n=
10
xi =
|xi - x
150
|2 = 446
Calculations :
i)
ii)
iii)
= 45%
Example Calculate s
...
of the marks of 100 students
...
of
Mid-
students
values
fi xi2
fi xi
102
(fi)
(xi)
0-2
10
1
10
10
2-4
20
3
60
180
4-6
35
5
175
875
6-8
30
7
210
1470
8-10
5
9
45
405
Sum fi xi =
Sum fi xi2 =
500
2940
n = 100
Solution
1)
2)
103
Example Calculate s
...
of the marks of 100 students
...
of
Mid-
students
values
(fi)
(xi)
0-2
10
2-4
n = 100
Solution
1)
2)
104
Example The score of two teams A and B in 10 matches are as:
A
40
32
0
40
30
7
13
25
14
3
B
21
14
14
30
5
12
10
13
30
6
Find the variance for both the series
...
4 Percentile
The nth percentile is that value ( or size ) such that n% of values of the whole data lies below it
...
Percentile Range
it is used as one of the measure of dispersion
...
The semi - percentile range, i
...
106
5
...
e
...
If we discard these two
values the limited range thus available might be more informative
...
It is the range which includes middle 50% of the distribution
...
Now the lower quartile ( Q1 ) is the 25th percentile and the upper quartile ( Q3 ) is the 75th
percentile
...
Thus symbolically
Inter quartile range = Q3 - Q1
If we divide ( Q3 - Q1 ) by 2 we get what is known as Semi-inter quartile range
...
e
...
It is known as Quartile deviation ( Q
...
2
Therefore Q
...
( SI QR ) =
Q3 Q1
2
107
SAMPLE CAT
1
...
The following is the distribution of weights of 140 students of the Business statistics class
of Mount Kenya University during the last intake
...
[3mks]
3
...
[2mks]
108
LECTURE 8
5
...
From the measures of variability, we can
know that whether most of the items of the data are close to our away from these central
tendencies
...
Another aspect of the data is to know its symmetry
...
This symmetry is well studied by the knowledge of the "skewness
...
This is understood by what
is known as “Kurtosis
...
For example,
see the following diagram
...
Where do they differ?
They differ in symmetry
...
For a symmetrical distribution, the
values, of equal distances on either side of the mode, have equal frequencies
...
Its curve rises slowly, reaches a maximum (peak) and falls
109
equally slowly (Fig
...
But for a skewed distribution, the mean, mode and median do not
coincide
...
A positively skewed distribution ( Fig
...
In other words, the tail as well as median on the right-hand side
...
3) rises slowly reaches its maximum and falls rapidly
...
5
...
1 Measure of Skewness
Pearson has suggested the use of this formula if it is not possible to determine the mode (Mo) of
any distribution,
Then,
SK
3mean median
s
...
ii) Sk = 0, then there is no skewness
iii) If Sk is positive, the skewness is also positive
...
110
1, but Karl Pearson’s co-
Unless and until no indication is given, you must use only Karl Pearson’s formula
...
Calculate the median income
...
of workers :
:
Below 50
1
50-70
16
70-90
39
90-110
58
110-130
60
130-150
46
150-170
22
170-190
15
190-210
15
210-230
9
230 & above
10
111
Solution:
Income
f
c
...
Below 50
1
1
50 – 70
16
17
70 – 90
39
56
90 – 110
58
114
110 – 130
60
174
130 – 150
46
220
150 – 170
22
242
170 – 190
15
257
190 – 210
15
252
210 – 230
9
281
230 &
10
291
group
above
n=
f = 291
Calculations :
1) Median = Size of
item
112
= Size of
item
= Size of 146th item which lies in (100-130) class interval
...
6
...
" In statistics it is the degree of flatness or
’peakedness’ in the region of mode of a frequency curve
...
It tells us the extent to which a distribution is more peaked or
flat-topped than the normal curve
...
’ In this case items are more clustered about the mode
...
The normal curve itself is known as "Meso
Kurtic
...
7 Characteristics of a good measure of Dispersion
i
...
ii
...
iii
...
The table shows the number of children in 50 families
...
(b)
Find the values of m, p and q
...
A group of 25 females were asked how many children they each had
...
(a)
Show that the mean number of children per female is 1
...
(2)
(b)
Show clearly that the standard deviation for this data is approximately 1
...
(3)
(c)
Another group of 25 females was surveyed and it was found that the mean
number of children per female was 2
...
Use
the results from parts (a) and (b) to describe the differences between the
number of children the two groups of females have
...
The following table shows the times, to the nearest minute, taken by 100 students to
complete a mathematics task
...
(Use upper class boundaries 15
...
5 and so on
...
(c)
Use your graph to estimate
(i)
the number of students that completed the task in less than 17
...
4
4
...
Marks
0–9
10–19 20–29 30–39 40–49 50–59 60–69 70–79 80–89
(%)
Frequenc
100
2
7
8
13
24
30
6
y
The following is the cumulative frequency table for the marks
...
5
2
< 19
...
5
s
< 39
...
5
54
< 59
...
5
t
116
5
3
2
(a)
< 79
...
5
98
< 100
100
Calculate the values of s and of t
...
(3)
(c)
Use your graph to estimate
(i)
the median mark;
(ii)
the lower quartile;
(iii) the pass mark, if 40% of the candidates passed
...
The cumulative frequency graph below shows the examination scores of 80
students
...
6
...
weight (kg)
frequency
cumulative frequency
0
...
70
16
16
0
...
90
37
53
0
...
10
44
c
118
(a)
(i)
1
...
30
23
120
1
...
50
10
130
Write down the value of c
...
Use a scale of 1 cm to represent 0
...
Label the axes clearly
...
95 kg
...
How many fish does the zoo buy?
(2)
(ii)
A pet food company buys all the fish in the lowest quartile
...
(i)
Calculate the minimum and maximum weights for the fish bought by
the restaurant
...
f
200
180
Number of students
160
140
120
100
80
60
40
20
0
(a)
5
10
15
20
25
30
35
Time in minutes
40
45
50
55
From the graph find
(i)
the median time;
(ii)
the interquartile range
...
Time in minutes
Number of students
0
20
5 < x ≤ 15
20
15 < x ≤ 20
p
20 < x ≤ 25
40
25 < x ≤ 35
60
35 < x ≤ 50
q
120
60
50 < x ≤ 60
(b)
10
Using the graph, find the values of p and q
...
References
Business Calculations and statistics simplified by N
...
Revised Edition
...
Objectives
a)
b)
c)
d)
Define correlation
Describe degrees of correlation
Identify methods of determining correlation
Compute Pearson’s correlation coefficient
...
6
...
By the averages, dispersion and
skewness of distribution, we get a complete idea about the structure of the distribution
...
If we carefully study the
figures of rain fall and production of maize, figures of accidents and motor cars in a city, of
demand and supply of a commodity, of sales and profit, we may find that there is some
relationship between the two variables
...
If there is any relation between two variables i
...
when one variable changes
the other also changes in the same or in the opposite direction, we say that the two variables are
correlated
...
2 Correlation Defined
122
It means the study of existence, magnitude and direction of the relation between two or more
variables
...
3 Types of Correlation
1
...
Linear and non-linear correlation
A) If two variables change in the same direction (i
...
if one increases the other also increases, or
if one decreases, the other also decreases), then this is called a positive correlation
...
B) If two variables change in the opposite direction ( i
...
if one increases, the other decreases and
vice versa), then the correlation is called a negative correlation
...
V
...
1
...
If the graph is in a straight line, the correlation is called a "linear correlation"
and if the graph is not in a straight line, the correlation is non-linear or curvi-linear
...
The ratio between the two always remains the same (1/5 in this case)
...
6
...
On the basis of the coefficient of correlation we can also determine
whether the correlation is positive or negative and also its degree or extent
...
Perfect correlation: If two variables changes in the same direction and in the same
proportion, the correlation between the two is perfect positive
...
On the other hand if the variables
change in the opposite direction and in the same proportion, the correlation is perfect
123
negative
...
In practice we rarely come across these types
of correlations
...
Absence of correlation: If two series of two variables exhibit no relations between them
or change in variable does not lead to a change in the other variable, then we can firmly
say that there is no correlation or absurd correlation between the two variables
...
Limited degrees of correlation: If two variables are not perfectly correlated or is there a
perfect absence of correlation, then we term the correlation as Limited correlation
...
High degree, moderate degree or low degrees are the three categories of this kind of correlation
...
Degrees
Positive
Negative
Absence of correlation
Zero
0
Perfect correlation
+1
-1
High degree
+ 0
...
25 to + 0
...
25
- 0
...
25 to - 0
...
25
6
...
124
6
...
1 Scatter Plot (Scatter diagram or dot diagram)
In this method the values of the two variables are plotted on a graph paper
...
By plotting the data, we get
points (dots) on the graph which are generally scattered and hence the name ‘Scatter Plot’
...
The degree of correlation is denoted by ‘ r ’ and its direction is given by the signs positive and
negative
...
1)
125
ii) If all points lie on a falling straight line the correlation is perfectly negative and r = -1 (see
fig
...
3)
iv) If the points lie in a narrow strip, falling downwards, the correlation is high degree of
negative (see fig
...
5)
vi) If the points are spread widely over a broad strip, falling downward, the correlation is low
degree negative (see fig
...
i
...
r
= 0
...
7)
Though this method is simple and is a rough idea about the existence and the degree of
correlation, it is not reliable
...
5
...
it is noted by ‘ r ’
...
Karl Pearson correlation
coefficient is also sometimes referred to as the product moment correlation coefficient and is
defined as
r
s xy
sx s y
Where
126
s xy
s xy
x x y y ,
n
x x
2
sx
n
y y
2
, sy
n
is called the covariance of X and Y
s is the standard deviation of X
x
s
y
is the standard deviation of Y
Therefore
r
x x y y
x x y y
2
2
Example 1
A chemical fertilizer company wishes to determine the extent of correlation between ‘quantity of
compound X used’ and ‘lawn growth’ per day
...
Solution
127
r
x x y y
x x y y
2
2
We start by obtaining the means x and y
...
969
10 18
A positive r means that as x (the mass of the chemical compound) increases, then so does y(the
lawn growth)
A value of r close to 1 indicates a very strong positive correlation
...
A second formula that does not require the calculation of the means is shown below
...
)
n
sx
x
x n
sy
y2
2
2
(An alternative formula for finding the standard deviation of X)
y 2
(An alternative formula for finding the standard deviation of Y)
n
Therefore,
r
xy
x
x2
2
n
x y
n
2
y
2
y
n
Using this formula, we can do example 1 above as follows:
x
y
x y
(x) 2
y 2
129
1
3
12
1
9
2
3
6
4
9
4
6
24
16
36
5
8
40
25
64
12
20
73
46
118
Therefore
12 20
4
r
144
400
46
118
4
4
13
10 18
0
...
a) s = 14
...
2, and
y
s xy = 136
...
8
0
...
7 19
...
986
Example 3
If covariance between x and y is 12
...
4 and 13
...
Find the coefficient of correlation between them
...
3 , s x 16
...
8
r
cov( XY )
sx s y
131
r
12
...
4 13
...
818
6
...
3 Spearman’s Rank Correlation Coefficient
This method is based on the ranks of the items rather than on their actual values
...
For example if you want to know the correlation between honesty and wisdom of the
boys of your class, you can use this method by giving ranks to the boys
...
The
formula is :
R 1
6 D 2
N N 2 1
where R = Rank correlation coefficient
D = Difference between the ranks of two items
N = The number of observations
...
iii)
When R = 0: No Correlation
...
Give ranks to the values of items
...
According to their values in the decreasing
order
...
Find the difference D = R1 - R2
where R1 = Rank of x and R2 = Rank of y
iii
...
Apply the formula
...
in such a case each items have ranks 4th
and 5th respectively then they are given
4th then they are given
factor i
45
= 4
...
If three items are of equal rank say
2
456
= 5th rank each
...
If there are more than one of such cases then this factor
12
added as many times as the number of such cases, then
133
Example Calculate ‘ R ’ from the following data
...
:
Rank
in
Maths :
Rank
in
Stats:
Solution :
Student
Rank
Rank
No
...
Marks
in Stats
40
42
45
35
36
39
46
43
44
39
40
43
:
Marks
in
English
:
135
Solution:
Marks
Marks
in
R1
in
R2
R1 - R2
(R1 -R2)2 =D2
English
Stats
40
3
46
1
2
4
42
2
43
3
...
5
2
...
5
0
...
25
N=6
D 0 D
2
7
...
D
2
is sometimes written as SD2
...
The sum of the squares of difference between the
corresponding rnarks was 55
...
Solution: We have
137
LECTURE 10
6
...
Now it is natural to think of a method that helps us in estimating the value of one
variable when the other is known
...
The fact that the
variables x and y are correlated does not necessarily mean that x causes y or vice versa
...
The reason for these accidents is not the school attendance; but these two
increases what is known as population
...
It assesses the contribution of one or more
variable called causing variable or independent variable or one which is being caused
(dependent variable)
...
This procedure is called simple linear regression
...
The dictionary meaning of
regression is "to go backward
...
"
6
...
if the strip is nearly straight, we can draw a straight line, such that all points are
close to it from both sides
...
This
line is called the line of best fit if it minimizes the distances of all data points from it
...
Now prediction is easy because now
all we need to do is to extend the line and read the value
...
But statisticians don’t measure the distances by dropping
perpendiculars from points on to the line
...
138
Thus we get two lines of regressions as shown in the figure (1) and (2)
...
6
...
The least squares formula is
yy
s xy
sx
2
x x
Example 1
Use the least squares formula to fit a regression line through (1,3), (3,5) and (5,6)
...
C
...
75 x 2
...
Year
1999
2000
2001
2002
2003
2004
Sales(Sh x1000)
5
9
14
18
21
27
a) Draw a scatter graph to represent this data
...
d) Predict the sales for the year 2006, giving your answer to the nearest Sh
Solution
a)
141
b) To get r2
y 2
x
y
x y
1
5
5
1
25
2
9
18
4
81
3
14
42
9
196
4
18
72
16
324
5
21
105
25
441
6
27
162
36
729
21
94
404
91
1796
r
r
=
xy
x
x2
2
n
404
212
91
6
(x) 2
x y
n
2
y
2
y
n
21 94
6
94 2
1796
6
404 329
91 73
...
66
75
75
...
997
142
r 2 0
...
5
6
yy
s xy
sx
2
x y
y 15
...
67
6
x x
s xy xy
s xy 404
y
,
x 91 21
2
n
6
2
17
...
5
17
...
67
17
...
y 4
...
67
In the year 2008, x 8
Therefore
143
y = (4
...
67
y= 34
...
67
Sales =34
...
Example3
A panel of two judges A and B graded dramatic performance by independently awarding marks
as follows:
a) Obtain the correlation coefficient r
b) Use the least squares method to obtain the regression equation of y on x
...
Solution:
a)
x
y
xy
x2
y2
36
35
1260
1296
1225
32
33
1056
1024
1089
34
31
1054
1156
961
31
30
930
961
900
32
34
1088
1024
1156
32
32
1024
1024
1024
144
35
36
1260
1225
1296
232
231
7672
7710
7651
Now
r
xy
x
x2
2
n
x y
n
2
y 2 y
n
232 231
7
r
2
232 7651 2312
7710
7
7
7672
7612 7656
7710 7689 7651 7623
16
21 28
Therefore r=0
...
76x+7
...
76 ( 38) +7
...
8 = 37 ( approximately )
Therefore, the Judge B would have given 37 marks to 8th performance
Alternative Formula for Calculating Regression
It is expressed as y = a + bx where a and b are two unknown constants which determine the
position of the line completely
...
The two basic equations which can be solved simultaneously to find a and b are:-
…………(i)
…
...
Estimate the supply when the price of
16 units
...
(2)
Solving (1) and (2) a
2860 = 130 a + 1690 b
3467 = 130 a + 2288 b
On subtraction,
607 = 598 b
b = 1
...
002 in 220 = 10 a + 130 b, we get a = 8
...
Hence the 3 equation of the line of regression of y on x is
y = 8
...
002 x
When x = 16, we get
y = 8
...
002 ( 16 )
y = 25
...
9 Uses of Regression Analysis
147
1
...
2
...
3
...
6
...
ii)
The cause and effect relation is clearly indicated through regression analysis but
we cannot say that one variable is the cause and the other the effect
...
Regression analysis has
much wider applications as it studies both linear and nonlinear relationship
between variables
...
148
Chapter Review Questions
1
...
Relationship between leaf length and width
70
60
50
Width
(mm) 40
30
20
10
0
(a)
(b)
20
40
60
80
100
Length (mm)
120
140
160
Draw a suitable line of best fit
...
2
...
I
High positive linear correlation
II Low positive linear correlation
III No correlation
IV Low negative linear correlation
V
High negative linear correlation
Which statement best represents the relationship between the two variables shown in
each of the scatter diagrams below
...
The Type Fast secretarial training agency has a new computer software
spreadsheet package
...
Fifteen
150
individuals are tested and the results are summarised in the table below
...
5 and Sxy = 36
...
(ii)
What does the value of the correlation coefficient suggest about the
relationship between the two variables?
(b)
Given that the mean time taken was 10
...
(c)
Use your equation for the regression line to predict
(i)
the time that it would take a 33 year old person to reach proficiency,
giving your answer correct to the nearest hour;
(ii)
the age of a person who would take 8 hours to reach proficiency, giving
your answer correct to the nearest year
...
Ten students were given two tests, one on Mathematics and one on English
...
151
Student
A
B
C
D
E
F
G
H
I
J
8
...
4 12
...
3
1
...
4 13
...
9 13
...
6
33
12
23
Mathematics
(x)
English
(y)
(a)
51
30
48
46
18
36
50
Given sxy (the covariance) is 35
...
(6)
(b)
Use your result from part (a) to comment on the statement:
'Those who do well in Mathematics also do well in English
...
The heights and weights of 10 students selected at random are shown
in the table below
...
Use a scale of 1 cm to represent 20
cm on the
x-axis and 1 cm to represent 10 kg on the y-axis
...
(c) Calculate the mean weight
...
31
...
276
...
(iii)
Draw the line of best fit on your graph
...
(f) It is decided to remove the data for student number 10 from all calculations
...
References
i
...
Pg132-134
ii
...
A Saleemi
...
Pg
480-501, 508-522
iii
...
Objectives
a) Define a random variable
...
Binomial distribution
ii
...
Normal distribution
c) Use tables to read probabilities for the above distributions
...
1 Introduction
Any variable can have a number of possible values
...
Each value (in case of a discrete variable) may have a particular
chance or probability of occurring
...
Recall ‘discrete’ means ‘positive whole number’
...
Example: We may have zero, one, two
...
The
number would be limited by the size of the room
...
Example : The number of people (X) in a moving car licensed to carry up to 6
people follows the pattern below: Number Probability
X
P (X)
1
0
...
3
154
3
0
...
05
5
0
...
02
1
...
1
Up to 3 people:
P (X ≤ 3) = 0
...
3 + 0
...
9
Fewer than 3 people: P (X < 3) = 0
...
3 = 0
...
05 + 0
...
02 =0
...
1 + 0
...
03 + 0
...
2
Between 2 and 5 people: P (2 ≤ X ≤ 5)* =
...
1 +
...
03 =
...
5 +
...
05 +
...
03 +
...
9
* P (2 ≤ X ≤ 5) is read as X is greater than or equal to 2 but less than or equal to 5
...
2 The Binomial Distribution
The binomial distribution is used when there are exactly two mutually exclusive outcomes of a
trial
...
The binomial distribution
is used to obtain the probability of observing x successes in N trials, with the probability of
success on a single trial denoted by p
...
155
7
...
1 Features of a Binomial Distribution
The Binomial experiment consists of:
A fixed number of trials, n
Two possible outcomes, p and q (or 1 – p)
...
‘Success’ is what we are interested in
...
p = 0
...
Independent trials (the outcome of one trial does not affect the outcomes of any other
trials)
...
2
...
Each page focuses on a given value of ‘n’
...
In
the left margin are the possible values of ‘k’
...
Example 1
Find the probability of at least two defective items in a batch of 10 items with a defective rate of
10%
...
156
Example 2:
A coin is tossed 10 times
...
5 and we require P(X ≤ 4)
Directly from the tables using k=4 we get P(X ≤ 4) = 0
...
What is the probability that there will be more than
9 girls?
Solution: n =15, p =
...
849
= 0
...
151 or
...
Example 3: The chance of any computer chip being defective is 20%
...
2
Reading from the Binomial tables,
a) P(X < 3)
= P(X ≤ 2)
= 0
...
035
157
c) P(X = 5)
= P(X ≤ 5) – P(X ≤ 4)
= 0
...
836
= 0
...
999- 0 939
= 0
...
939
Alternatively, we could have used n=15, p= 0
...
061
= 0
...
Note: *The designations for success and failure can be interchanged
So far we have shown how we determine the probabilities of a binomial experiment
by using the Binomial Tables
...
Select ‘statistical’ and ‘binomdist’)
...
You will use the tables
...
x represents the value of the random
variable
...
6! means 6 x 5 x 4 x 3 x
2 x 1 = 720
...
7
...
What is the probability that it will occur k times in that period?
A Poisson probability distribution is the number of occurrences per interval of time or
space
...
Space?
Example: A programmer makes, on average, 2 mistakes per program
...
What is
the probability that in a given hour there will be 5 arrivals? Here we have μ=3 and
k=5
...
7
...
1 Features of the Poisson Experiment
The number of successes that occur in a period of time or an interval of space is
independent of the number of successes that occur in any other interval
159
The probability of a success in an interval is the same for all equal-sized intervals and
proportional to the size of the interval (for example an average of 4 in an hour is
equivalent to an average of 2 in 30 minutes (1/2 hour) and 8 in two hours)
...
2) The stated mean applies only to the given specific situation or time period
...
Example: If 4 trucks cross a bridge in 20 minutes (on average), then 2 trucks will cross the
bridge in a 10 minute period (on average)
...
3
...
In the left margin are the possible values of
‘k’
...
Eg
...
916
...
Illustration : Poisson Tables for μ values are at the back of this module
...
What is the probability that a TV set chosen at
random will have a) not more than 4 faults? b) more than 6 faults? c) between 3 and 5 faults? d)
exactly 2 faults?
Solutions : For all cases μ = 3
...
815
160
b) P(X > 6) = P(X ≥ 7) = 1 – P(X ≤ 6)
= 1 - 0
...
034
c) P(3 ≤ X ≤ 5) = P(X ≤ 5) – P(X ≤ 2)
= 0
...
423
= 0
...
423 - 0
...
244
Example 2
The average number of telephone calls you receive in your office every
hour is 3
...
What is the probability that you will not miss any calls during that time?
b) In fact the ‘urgent business’ takes two hours
...
a) μ = 1
...
223
b) μ = 6
...
285
= 0
...
5
...
5, we need to find a value of k such that P(X ≤ k) >
...
758
4
...
958
Five crews are needed to cover at least 95% of emergencies
...
Normally the question asks
for a probability and gives the x or k value
...
So far we have shown how we determine the probabilities of a Poisson experiment
By using the Poisson Tables
...
However, you are not required to use formula in
this course
...
For your information only the formula is:
X=
e is the base of the natural logarithm (approximately 2
...
162
7
...
3 Conditions under which a Poisson distribution holds
counts of rare events
all events are independent
average rate does not change over the period of interest
7
...
4 Examples of experiments where a Poisson distribution holds:
birth defects
number of sample defects on a car
number of typographical errors on a page
Examples of Poisson probability distribution:
The mass of alpha particles released by a radioactive source in a known interval of time
...
The amount of imperfect research paper in a packet of 100, created by a good industry
...
The number of road accidents reports in a city at a particular junction at a particular tim
7
...
5 Examples of experiments where a Poisson distribution may not hold
number of insects on a tree - contagion?
number of males in families of size 4 - not `rare' events
163
LECTURE 12
7
...
It is known as a normal random variable, and its probability distribution is called a
normal distribution
...
4
...
It is bell shaped and is symmetrical about its mean
...
It is asymptotic to the axis, i
...
, it extends indefinitely in either direction from the mean
...
It is a continuous distribution
...
It is a family of curves, i
...
, every unique pair of mean and standard deviation defines a
different normal distribution
...
See the following figure
...
Total area under the curve sums to 1, i
...
, the area of the distribution on each side of the mean
is 0
...
6
...
e
...
7
...
7
...
However, this can be avoided by transforming all normal distribution to fit the standard normal
distribution
...
) to a standard measure called Z score or Z value
...
If the value of X is greater
than the mean, the Z score is positive; if the value of X is less than the mean, the Z score is
negative
...
The Z distribution is a normal distribution with a mean of 0 and a standard
deviation of 1
...
Graph the normal distribution, and shade the area related to the probability you want to find
...
Convert the boundaries of the shaded area from X values to the standard normal random
variable Z values using the Z formula above
...
Use the standard Z table to find the probabilities or the areas related to the Z values in step 2
...
Suppose that in one particular year, the mean score for the
GMAT was 476, with a standard deviation of 107
...
What is the probability that a randomly selected score from this GMAT falls
between 476 and 650? <= x <="650)" the following figure shows a graphic representation of this
problem
...
62
...
62 indicates that
the GMAT score of 650 is 1
...
The standard normal table
gives the probability of value falling between 650 and the mean
...
Across the top of the table are
the values of the hundredths place portion of the Z score
...
4474 or
44
...
Question 2
...
e
...
The Z score
is: Z = ( 750 - 476)/107 = 2
...
From the table, the probability for this Z score is 0
...
This is
the probability of a GMAT with a score between 476 and 750
...
50
...
5 - 0
...
0052 or 0
...
Note that P(X >= 750) is the same as P(X
>750), because, in continuous distribution, the area under an exact number such as X=750 is
zero
...
166
Figure 5
Question 3
...
e
...
" we are asked to determine
the area under the curve for all values less than or equal to 540
...
6
...
2257 which is the probability
of getting a score between the mean (476) and 540
...
Thus, the answer to this problem is: 0
...
2257 = 0
...
The following figure
shows a graphic representation of this problem
...
What is the probability of receiving a score between 440 and 330 on a GMAT test
that has a mean of 476 and a standard deviation of 107? i
...
, P(330 < 440)="?
...
Figure 7
In this problem, the two values fall on the same side of the mean
...
36, and Z2 = (440 - 476)/107 = -0
...
The probability associated with Z = -1
...
4131, and the probability associated with Z = -0
...
1331
...
Thus, the answer
to this problem is: 0
...
1331 = 0
...
Example 2:
Suppose that a tire factory wants to set a mileage guarantee on its new model called LA 50 tire
...
The factory wants to set the guaranteed
mileage so that no more than 5% of the tires will have to be replaced
...
e
...
deviation are given, but X and Z are unknown
...
05 of the X values less than that value
...
05 of the values are less than X, then 0
...
5 - 0
...
168
Refer to the standard normal distribution table and search the body of the table for 0
...
Since
the exact number is not found in the table, search for the closest number to 0
...
There are two
values equidistant from 0
...
4505 and 0
...
Move to the left from these values, and read
the Z scores in the margin, which are: 1
...
64
...
e
...
65 + 1
...
645
...
645 =(X - 47,900)/2,050 = 44,528 miles
...
169
LECTURE 13
7
...
You must look up the
area between zero and the value on the inside part of the table, and then read the z-score
from the outside
...
Remember, z-scores can be
negative, but areas or probabilities cannot be
...
5000
Look up the difference in the table
Make negative if in the left tail
Area including one complete half
Subtract 0
...
000
(More than z units from the mean)
Divide the area by 2
Look up the quotient in the table
Use both the positive and negative z-scores
Using the table becomes proficient with practice, work lots of the normal probability problems!
170
-3
-2
-1
0
1
2
3
Z
The shaded area between the middle (Z =0) and any value of Z is given in the
tables
...
62
The area from a Z of 0 to a Z of 1
...
4474
...
Example : The heights of adult males is normally distributed with mean 170cm and
standard deviation 10cm
...
Find the probability of a male between 180 and 190 cm
...
b) Use formula
Note: Strictly speaking a random variable is usually denoted by an uppercase X
whereas a lower case x represents one of its values
...
To calculate Z values of the boundaries of shaded area
Find areas in tables
Z
Area
1
...
4772
d) Add or subtract table areas to get probability required
...
4772 - 0
...
1359
Q2
...
4772
P(X>190) = 0
...
4772
= 0
...
Shorter than 180cm
Z= 180- 170 =1
10
Tables:
...
5 +0
...
8413
Q4
...
Tables :0
...
5
10
P(X<165) = 0
...
1915
= 0
...
List 5 characteristics of the normal curve
...
With the aid of clearly labeled normal curves, find the normal curve between
i) Z=-2 and Z=2
...
4
iii)To the left of z=2
...
3 and Z=-1
...
The marks obtained in a Business statistics CAT are normally distributed with mean 23 and
standard deviation 4
...
Find the probability that a randomly selected student scores
i) More than 25 marks
ii) Between 20 and25 marks
174
iii)Less than the mean mark?
4
...
If 10 dry cells are
selected at random, what is the probability that
a) Fewer than 3 will be defective?
b) None will be defective?
c) Exactly 5 will be defective?
d) Between 6 and 8 will be defective?
e) At least 10 will not be defective?
5
...
What is
the probability that in one minute,
i) No fewer than 5 cars pass through the junction
ii) Exactly 12 cars pass through the junction
iii) Between 5 and 7 cars pass through the junction?
Reference
i
...
Quantitative Methods for Business by Donald Waters Pg382-393
175
SAMPLE PAPERS
TEST PAPER 1
MT KENYA UNIVERSITY
UNIVERSITY EXAMINATIONS APRIL 2010
FIRST YEAR SEMESTER I EXAMINATION FOR THE DEGREE OF BACHELOR OF
COMMERCE AND BACHELOR OF BUSINESS MANAGEMENT
SBC223 / BBM223
INTRODUCTION TO BUSINESS STATISTICS
DATE: APRIL 2010
TIME: 3 HOURS
Instructions: Answer Question One and Any Other Two Questions
QUESTION ONE (Compulsory)
(30 marks)
(a) Using a well labeled diagram, show the position of the mean, mode and the median in
a negatively skewed distribution
(3marks)
(b) (b) The weighted mean of the two numbers 30 and 15 is 20
...
(3marks)
(c) For a skewed distribution, the mean is 86, the median is 20 and the standard
deviation is 5
...
(3marks)
(e) Given a set of data; 2,9,8,4,7,6
i)
Calculate the arithmetic mean
(2 marks)
ii)
Calculate the geometric mean
(2 marks)
iii)
Calculate the harmonic mean
(2 marks)
iv)
State the median
(1 mark )
v)
Calculate the standard deviation
...
A random sample of
100 cartons gave the following results for the volume, x
...
4 , x
2
102
...
Calculate the mean and standard deviation of the volume of juice in those 100
cartons
...
ii) Level of satisfaction with a meal at a family restaurant
...
(1mark)
iv) The weight of dog food
...
(1mark)
177
QUESTION TWO
(20 marks)
The manager of a fast food restaurant is concerned that the customers are waiting for too long
for their food
...
1
...
5
8
...
6 10
...
9
3
...
6
8
...
3
1
...
7
11
...
4
4
...
4
3
...
25
7
...
1 5
...
95 7
...
7 10
...
5
5
...
1
2
...
4
6
...
3
7
...
6 3
...
55 2
...
8
2
...
1
8
...
1
3
...
1
5
...
5 6
...
4 4
...
6
Group this
(a)
data into classes of 0 - 1
...
9 etc and construct a frequency table which also
shows cumulative frequencies
...
(4marks)
(c) Construct a less than ogive for the data and answer the following questions
(4marks)
i)
How many customers have to wait for less than 4 minutes to be served?
(2marks)
ii)
What percentage of customers has to wait for less than 5 minutes for their food?
(2marks)
iii)
iv)
If the restaurant’s goal is for 90% of the customers to be given their food within 8
minutes, are they achieving this goal?
(2marks)
What is the mean waiting time?
2marks)
178
QUESTION THREE
(20 marks)
a) A random sample of 51 people was asked to record the number of miles they travelled by
car in a given week
...
42
93
46
52
72
77
53
41
48
86
62
54
85
60
58
43
58
43
58
74
52
82
78
86
94
63
72
63
72
44
78
56
80
44
52
74
68
82
57
47
a) Construct a stem and leaf diagram to represent these data
...
(4marks)
c) Draw a box plot to represent these data
...
b) At the end of a statistics course, Diana sits for two written papers, P1 and P2 and hands in
a piece of course work
...
Her overall percentage is to be weighted so that the two
written papers account for 40% while the course work accounts for 20%
...
(4 marks)
179
QUESTION FOUR
(20 marks)
a) A reaction time experiment was performed first with 21 girls, and then with 24
boys
...
18
6|1 means0
...
(2marks)
180
b) In a practical class, students timed how long it took for a sample of their saliva
to break down a 2% starch solution
...
(4marks)
ii) Calculate the inter quartile range
(3marks)
iii) Calculate the coefficient of skewness
(3marks)
iv) Comment on the skewness of this distribution
(2marks)
QUESTION FOUR
(20 marks)
c) A random sample of 51 people was asked to record the number of miles they travelled by
car in a given week
...
181
42
93
46
52
72
77
53
41
48
86
62
54
85
60
58
43
58
43
58
74
52
82
78
86
94
63
72
63
72
44
78
56
80
44
52
74
68
82
57
47
e) Construct a stem and leaf diagram to represent these data
...
(4marks)
g) Draw a box plot to represent these data
...
d) At the end of a statistics course, Diana sits for two written papers, P1 and P2 and hands in
a piece of course work
...
Her overall percentage is to be weighted so that the two
written papers account for 40% while the course work accounts for 20%
...
(4 marks)
QUESTION FIVE (20 Marks)
a) A shopkeeper wanted to investigate whether or not there was a correlation
between the prices of food 10 years ago in 1992, with their prices today
...
182
1992 price
sugar
milk
eggs
rolls
tea bags
coffee
potatoes
flour
$
$ 0
...
16
$ 1
...
92
$ 3
...
32
$ 1
...
04
$ 2
...
00
$ 1
...
28
$ 1
...
44
1
...
20
(a)
Calculate the mean and the standard deviation of the prices
(i)
in 1992;
(ii)
in 2002
...
3104, calculate the correlation coefficient
...
(4)
(c)
Find the equation of the line of the best fit in the form y = mx + c
...
60 in 1992?
(1)
(e)
Which item would you omit to increase the correlation coefficient?
(2)
b) For a skewed distribution, the mean is 86, the median is 20 and the standard deviation is
5
...
(2)
c) Briefly explain any two instances in which the knowledge of Business Statistics may be
applied in making a managerial decision
...
(2 marks)
c) State Three reasons why a researcher may prefer to study a sample instead of the whole
population
...
(3 marks)
e) Using a clearly labeled diagram, show the position of the mode, mean and median of a
positively skewed distribution
...
(3marks)
ii) Comment on the skewness of the data involved
...
(3marks)
h) Equity Bank Ltd is studying the number of times their automatic teller machine located at
Tom Mboya Street is used daily
...
83 64 84 76 54 84 75 59 70 61
63 83 84 70 68 52 65 90 52 77
95 36 78 61 59 84 95 87 47 60
Construct a stem and leaf chart to represent the information above
...
Amt of milk(ltrs) 6–10 10 –14 14 –18 18–22 22–26 26–30 30 –34
No
...
The table below shows some results
...
(2 marks)
iii) Passed given that he/she was absent for less than four days
...
(2 marks)
d) Differentiate between nominal and ordinal scales of measurement
...
(4 marks)
b) The variation of incomes of executives is to be compared with the variation of incomes of
unskilled employees
...
For a sample of unskilled employees, the
mean = Ksh 12,000 and standard deviation = Ksh 2000
...
(5 marks)
c) At the end of a statistics course, Diana sits for two written papers, P1 and P2 and hands in
a piece of course work
...
Her overall percentage is to be weighted so that the two
written papers account for 40% while the course work accounts for 20%
...
(4 marks)
d) What is a discrete variable?
(2 marks
e) Dispersion is a statistical name for spread or variability while skewness describes how
non-symmetric the data is
...
187
(2 marks)
(3 marks)
QUESTION FOUR
(20 marks)
a) Marks out of 100 for 100 students were tabulated as shown below:
Marks
f
11 – 20
4
21 – 30
16
31 – 40
27
41 – 50
32
51 – 60
15
61 – 70
4
71 – 80
2
i) Calculate the mean mark
...
(4 marks)
iii) Use the ogive to estimate the upper quartile and the lower quartile
(2 marks)
iii)
Estimate from your ogive the minimum marks required to get grade A if
iv)
only 5 students are to get grade A
...
The class calculate the mean
height to be x = 12
...
35 m
...
5 m and 43
...
(a)
How many standard deviations away from the mean of 12
...
5?
(3marks)
188
The incorrect measurements of 44
...
2 m must be removed from the data
...
(3 marks)
QUESTION FIVE
(20marks)
The following data relates to daily bill on consumption of a certain commodity for 60
households
Daily bills(KSh)
10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No
...
NORMAL DISTRIBUTION TABLES
2
...
POISSON TABLES
Standard Normal Probabilities
z
0
...
01
0
...
03
0
...
05
0
...
07
0
...
09
0
...
0000 0
...
0080 0
...
0160 0
...
0239 0
...
0319 0
...
1 0
...
0438 0
...
0517 0
...
0596 0
...
0675 0
...
0753
0
...
0793 0
...
0871 0
...
0948 0
...
1026 0
...
1103 0
...
3 0
...
1217 0
...
1293 0
...
1368 0
...
1443 0
...
1517
0
...
1554 0
...
1628 0
...
1700 0
...
1772 0
...
1844 0
...
5 0
...
1950 0
...
2019 0
...
2088 0
...
2157 0
...
2224
0
...
2257 0
...
2324 0
...
2389 0
...
2454 0
...
2517 0
...
7 0
...
2611 0
...
2673 0
...
2734 0
...
2794 0
...
2852
0
...
2881 0
...
2939 0
...
2995 0
...
3051 0
...
3106 0
...
9 0
...
3186 0
...
3238 0
...
3289 0
...
3340 0
...
3389
1
...
3413 0
...
3461 0
...
3508 0
...
3554 0
...
3599 0
...
1 0
...
3665 0
...
3708 0
...
3749 0
...
3790 0
...
3830
1
...
3849 0
...
3888 0
...
3925 0
...
3962 0
...
3997 0
...
3 0
...
4049 0
...
4082 0
...
4115 0
...
4147 0
...
4177
1
...
4192 0
...
4222 0
...
4251 0
...
4279 0
...
4306 0
...
5 0
...
4345 0
...
4370 0
...
4394 0
...
4418 0
...
4441
1
...
4452 0
...
4474 0
...
4495 0
...
4515 0
...
4535 0
...
7 0
...
4564 0
...
4582 0
...
4599 0
...
4616 0
...
4633
190
1
...
4641 0
...
4656 0
...
4671 0
...
4686 0
...
4699 0
...
9 0
...
4719 0
...
4732 0
...
4744 0
...
4756 0
...
4767
2
...
4772 0
...
4783 0
...
4793 0
...
4803 0
...
4812 0
...
1 0
...
4826 0
...
4834 0
...
4842 0
...
4850 0
...
4857
2
...
4861 0
...
4868 0
...
4875 0
...
4881 0
...
4887 0
...
3 0
...
4896 0
...
4901 0
...
4906 0
...
4911 0
...
4916
2
...
4918 0
...
4922 0
...
4927 0
...
4931 0
...
4934 0
...
5 0
...
4940 0
...
4943 0
...
4946 0
...
4949 0
...
4952
2
...
4953 0
...
4956 0
...
4959 0
...
4961 0
...
4963 0
...
7 0
...
4966 0
...
4968 0
...
4970 0
...
4972 0
...
4974
2
...
4974 0
...
4976 0
...
4977 0
...
4979 0
...
4980 0
...
9 0
...
4982 0
...
4983 0
...
4984 0
...
4985 0
...
4986
3
...
4987 0
...
4987 0
...
4988 0
...
4989 0
...
4990 0
...
That is, P(0
191
Binomial Distribution Tables
N=2
K \ P=
...
2
...
4
...
6
...
8
...
81 0
...
49 0
...
25 0
...
09 0
...
01
1 | 0
...
96 0
...
84 0
...
64 0
...
36 0
...
00 1
...
00 1
...
00 1
...
00 1
...
00
N=3
K \ P=
...
2
...
4
...
6
...
8
...
729 0
...
343 0
...
125 0
...
027 0
...
001
1 | 0
...
896 0
...
648 0
...
352 0
...
104 0
...
999 0
...
973 0
...
875 0
...
657 0
...
271
3 | 1
...
000 1
...
000 1
...
000 1
...
000 1
...
1
...
3
...
5
...
7
...
9
-----------------------------------------------------------------0 | 0
...
4096 0
...
1296 0
...
0256 0
...
0016 0
...
9477 0
...
6517 0
...
3125 0
...
0837 0
...
0037
2 | 0
...
9728 0
...
8208 0
...
5248 0
...
1808 0
...
9999 0
...
9919 0
...
9375 0
...
7599 0
...
3439
4 | 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
1
...
3
...
5
...
7
...
9
--------------------------------------------------------------------------0 | 0
...
32768 0
...
07776 0
...
01024 0
...
00032 0
...
91854 0
...
52822 0
...
18750 0
...
03078 0
...
00046
2 | 0
...
94208 0
...
68256 0
...
31744 0
...
05792 0
...
99954 0
...
96922 0
...
81250 0
...
47178 0
...
08146
4 | 0
...
99968 0
...
98976 0
...
92224 0
...
67232 0
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
00000
N=6
K \ P=
...
2
...
4
...
6
...
8
...
53144 0
...
11765 0
...
01562 0
...
00073 0
...
00000
1 | 0
...
65536 0
...
23328 0
...
04096 0
...
00160 0
...
98415 0
...
74431 0
...
34375 0
...
07047 0
...
00127
192
3 | 0
...
98304 0
...
82080 0
...
45568 0
...
09888 0
...
99994 0
...
98906 0
...
89062 0
...
57982 0
...
11426
5 | 1
...
99994 0
...
99590 0
...
95334 0
...
73786 0
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
00000
N=7
K \ P=
...
2
...
4
...
6
...
8
...
47830 0
...
08235 0
...
00781 0
...
00022 0
...
00000
1 | 0
...
57672 0
...
15863 0
...
01884 0
...
00037 0
...
97431 0
...
64707 0
...
22656 0
...
02880 0
...
00018
3 | 0
...
96666 0
...
71021 0
...
28979 0
...
03334 0
...
99982 0
...
97120 0
...
77344 0
...
35293 0
...
02569
5 | 0
...
99963 0
...
98116 0
...
84137 0
...
42328 0
...
00000 0
...
99978 0
...
99219 0
...
91765 0
...
52170
7 | 1
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
1
...
3
...
5
...
7
...
9
--------------------------------------------------------------------------0 | 0
...
16777 0
...
01680 0
...
00066 0
...
00000 0
...
81310 0
...
25530 0
...
03516 0
...
00129 0
...
00000
2 | 0
...
79692 0
...
31539 0
...
04981 0
...
00123 0
...
99498 0
...
80590 0
...
36328 0
...
05797 0
...
00043
4 | 0
...
98959 0
...
82633 0
...
40591 0
...
05628 0
...
99998 0
...
98871 0
...
85547 0
...
44823 0
...
03809
6 | 1
...
99992 0
...
99148 0
...
89362 0
...
49668 0
...
00000 1
...
99993 0
...
99609 0
...
94235 0
...
56953
8 | 1
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
1
...
3
...
5
...
7
...
9
--------------------------------------------------------------------------0 | 0
...
13422 0
...
01008 0
...
00026 0
...
00000 0
...
77484 0
...
19600 0
...
01953 0
...
00043 0
...
00000
2 | 0
...
73820 0
...
23179 0
...
02503 0
...
00031 0
...
99167 0
...
72966 0
...
25391 0
...
02529 0
...
00006
4 | 0
...
98042 0
...
73343 0
...
26657 0
...
01958 0
...
99994 0
...
97471 0
...
74609 0
...
27034 0
...
00833
6 | 1
...
99969 0
...
97497 0
...
76821 0
...
26180 0
...
00000 0
...
99957 0
...
98047 0
...
80400 0
...
22516
8 | 1
...
00000 0
...
99974 0
...
98992 0
...
86578 0
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
00000
193
N =10
K \ P=
...
2
...
4
...
6
...
8
...
34868 0
...
02825 0
...
00098 0
...
00001 0
...
00000
1 | 0
...
37581 0
...
04636 0
...
00168 0
...
00000 0
...
92981 0
...
38278 0
...
05469 0
...
00159 0
...
00000
3 | 0
...
87913 0
...
38228 0
...
05476 0
...
00086 0
...
99837 0
...
84973 0
...
37695 0
...
04735 0
...
00015
5 | 0
...
99363 0
...
83376 0
...
36690 0
...
03279 0
...
99999 0
...
98941 0
...
82812 0
...
35039 0
...
01280
7 | 1
...
99992 0
...
98771 0
...
83271 0
...
32220 0
...
00000 1
...
99986 0
...
98926 0
...
85069 0
...
26390
9 | 1
...
00000 0
...
99990 0
...
99395 0
...
89263 0
...
00000 1
...
00000 1
...
00000 1
...
00000 1
...
00000
Cumulative Binomial Distribution Table
n=1
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
9800 0
...
9600 0
...
9400 0
...
9200 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
15
p=
0
...
25
p=
0
...
35
p=
0
...
45
p=
0
...
9000 0
...
8000 0
...
7000 0
...
6000 0
...
5000
1
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
01
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
4500 0
...
3500 0
...
2500 0
...
1500 0
...
0900
1
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0700 0
...
0500 0
...
0300 0
...
0100 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=2
x
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
9604 0
...
9216 0
...
8836 0
...
8464 0
...
9999 0
...
9991 0
...
9975 0
...
9951 0
...
9919
2
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
0
...
20
0
...
30
0
...
40
0
...
50
0
0
...
7225 0
...
5625 0
...
4225 0
...
3025 0
...
9900 0
...
9600 0
...
9100 0
...
8400 0
...
7500
2
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
55
p=
0
...
65
p=
0
...
75
p=
0
...
85
p=
0
...
91
0
0
...
1600 0
...
0900 0
...
0400 0
...
0100 0
...
6975 0
...
5775 0
...
4375 0
...
2775 0
...
1719
2
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
p=
0
...
93
p=
0
...
95
p=
0
...
97
p=
0
...
99
p=
1
...
0064 0
...
0036 0
...
0016 0
...
0004 0
...
0000
1
0
...
1351 0
...
0975 0
...
0591 0
...
0199 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=3
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
9412 0
...
8847 0
...
8306 0
...
7787 0
...
9997 0
...
9974 0
...
9928 0
...
9860 0
...
9772
2
1
...
0000 1
...
9999 0
...
9998 0
...
9995 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
15
p=
0
...
25
p=
0
...
35
p=
0
...
45
p=
0
...
7290 0
...
5120 0
...
3430 0
...
2160 0
...
1250
1
0
...
9393 0
...
8438 0
...
7183 0
...
5748 0
...
9990 0
...
9920 0
...
9730 0
...
9360 0
...
8750
3
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
55
p=
0
...
65
p=
0
...
75
p=
0
...
85
p=
0
...
91
0
0
...
0640 0
...
0270 0
...
0080 0
...
0010 0
...
4253 0
...
2818 0
...
1563 0
...
0608 0
...
0228
2
0
...
7840 0
...
6570 0
...
4880 0
...
2710 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0003 0
...
0001 0
...
0000 0
...
0000 0
...
0182 0
...
0104 0
...
0047 0
...
0012 0
...
0000
196
2
0
...
1956 0
...
1426 0
...
0873 0
...
0297 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=4
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
9224 0
...
8493 0
...
7807 0
...
7164 0
...
9994 0
...
9948 0
...
9860 0
...
9733 0
...
9570
2
1
...
0000 0
...
9998 0
...
9992 0
...
9981 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
9999
4
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
5220 0
...
3164 0
...
1785 0
...
0915 0
...
9477 0
...
8192 0
...
6517 0
...
4752 0
...
3125
2
0
...
9880 0
...
9492 0
...
8735 0
...
7585 0
...
9999 0
...
9984 0
...
9919 0
...
9744 0
...
9375
4
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
01
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
0410 0
...
0150 0
...
0039 0
...
0005 0
...
0001
1
0
...
1792 0
...
0837 0
...
0272 0
...
0037 0
...
6090 0
...
4370 0
...
2617 0
...
1095 0
...
0430
3
0
...
8704 0
...
7599 0
...
5904 0
...
3439 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
197
x
p=
0
...
93
p=
0
...
95
p=
0
...
97
p=
0
...
99
p=
1
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
1
0
...
0013 0
...
0005 0
...
0001 0
...
0000 0
...
0344 0
...
0199 0
...
0091 0
...
0023 0
...
0000
3
0
...
2519 0
...
1855 0
...
1147 0
...
0394 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=5
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
9039 0
...
8154 0
...
7339 0
...
6591 0
...
9990 0
...
9915 0
...
9774 0
...
9575 0
...
9326
2
1
...
9999 0
...
9994 0
...
9980 0
...
9955 0
...
0000 1
...
0000 1
...
0000 0
...
9999 0
...
9997
4
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
10
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
4437 0
...
2373 0
...
1160 0
...
0503 0
...
9185 0
...
7373 0
...
5282 0
...
3370 0
...
1875
2
0
...
9734 0
...
8965 0
...
7648 0
...
5931 0
...
9995 0
...
9933 0
...
9692 0
...
9130 0
...
8125
4
1
...
9999 0
...
9990 0
...
9947 0
...
9815 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
198
x
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
0185 0
...
0053 0
...
0010 0
...
0001 0
...
0000
1
0
...
0870 0
...
0308 0
...
0067 0
...
0005 0
...
4069 0
...
2352 0
...
1035 0
...
0266 0
...
0063
3
0
...
6630 0
...
4718 0
...
2627 0
...
0815 0
...
9497 0
...
8840 0
...
7627 0
...
5563 0
...
3760
5
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0002 0
...
0001 0
...
0000 0
...
0000 0
...
0000
2
0
...
0031 0
...
0012 0
...
0003 0
...
0000 0
...
0544 0
...
0319 0
...
0148 0
...
0038 0
...
0000
4
0
...
3043 0
...
2262 0
...
1413 0
...
0490 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=6
x
p=
0
...
02
p=
0
...
04
p=
0
...
06
p=
0
...
08
p=
0
...
9415 0
...
8330 0
...
7351 0
...
6470 0
...
5679
1
0
...
9943 0
...
9784 0
...
9541 0
...
9227 0
...
0000 0
...
9995 0
...
9978 0
...
9942 0
...
9882
3
1
...
0000 1
...
0000 0
...
9998 0
...
9995 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
199
5
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
3771 0
...
1780 0
...
0754 0
...
0277 0
...
8857 0
...
6554 0
...
4202 0
...
2333 0
...
1094
2
0
...
9527 0
...
8306 0
...
6471 0
...
4415 0
...
9987 0
...
9830 0
...
9295 0
...
8208 0
...
6563
4
0
...
9996 0
...
9954 0
...
9777 0
...
9308 0
...
0000 1
...
9999 0
...
9993 0
...
9959 0
...
9844
6
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
55
p=
0
...
65
p=
0
...
75
p=
0
...
85
p=
0
...
91
0
0
...
0041 0
...
0007 0
...
0001 0
...
0000 0
...
0692 0
...
0223 0
...
0046 0
...
0004 0
...
0000
2
0
...
1792 0
...
0705 0
...
0170 0
...
0013 0
...
5585 0
...
3529 0
...
1694 0
...
0473 0
...
0118
4
0
...
7667 0
...
5798 0
...
3446 0
...
1143 0
...
9723 0
...
9246 0
...
8220 0
...
6229 0
...
4321
6
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
p=
0
...
93
p=
0
...
95
p=
0
...
97
p=
0
...
99
p=
1
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
1
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0005 0
...
0002 0
...
0000 0
...
0000 0
...
0000
3
0
...
0058 0
...
0022 0
...
0005 0
...
0000 0
...
0773 0
...
0459 0
...
0216 0
...
0057 0
...
0000
5
0
...
3530 0
...
2649 0
...
1670 0
...
0585 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=7
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
8681 0
...
7514 0
...
6485 0
...
5578 0
...
9980 0
...
9829 0
...
9556 0
...
9187 0
...
8745
2
1
...
9997 0
...
9980 0
...
9937 0
...
9860 0
...
0000 1
...
0000 0
...
9998 0
...
9993 0
...
9982
4
1
...
0000 1
...
0000 1
...
0000 1
...
9999 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
6
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
10
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
3206 0
...
1335 0
...
0490 0
...
0152 0
...
8503 0
...
5767 0
...
3294 0
...
1586 0
...
0625
2
0
...
9262 0
...
7564 0
...
5323 0
...
3164 0
...
9973 0
...
9667 0
...
8740 0
...
7102 0
...
5000
4
0
...
9988 0
...
9871 0
...
9444 0
...
8471 0
...
0000 0
...
9996 0
...
9962 0
...
9812 0
...
9375
6
1
...
0000 1
...
9999 0
...
9994 0
...
9963 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
0037 0
...
0006 0
...
0001 0
...
0000 0
...
0000
1
0
...
0188 0
...
0038 0
...
0004 0
...
0000 0
...
1529 0
...
0556 0
...
0129 0
...
0012 0
...
0001
3
0
...
2898 0
...
1260 0
...
0333 0
...
0027 0
...
6836 0
...
4677 0
...
2436 0
...
0738 0
...
0193
5
0
...
8414 0
...
6706 0
...
4233 0
...
1497 0
...
9848 0
...
9510 0
...
8665 0
...
6794 0
...
4832
7
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
2
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0012 0
...
0004 0
...
0001 0
...
0000 0
...
0000
4
0
...
0097 0
...
0038 0
...
0009 0
...
0000 0
...
1026 0
...
0618 0
...
0294 0
...
0079 0
...
0000
6
0
...
3983 0
...
3017 0
...
1920 0
...
0679 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=8
x
p=
0
...
02
p=
0
...
04
p=
0
...
06
p=
0
...
08
p=
0
...
9227 0
...
7837 0
...
6634 0
...
5596 0
...
4703
1
0
...
9897 0
...
9619 0
...
9208 0
...
8702 0
...
9999 0
...
9987 0
...
9942 0
...
9853 0
...
9711
3
1
...
0000 0
...
9998 0
...
9993 0
...
9978 0
...
0000 1
...
0000 1
...
0000 1
...
9999 0
...
9997
5
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
7
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
2725 0
...
1001 0
...
0319 0
...
0084 0
...
8131 0
...
5033 0
...
2553 0
...
1064 0
...
0352
2
0
...
8948 0
...
6785 0
...
4278 0
...
2201 0
...
9950 0
...
9437 0
...
8059 0
...
5941 0
...
3633
4
0
...
9971 0
...
9727 0
...
8939 0
...
7396 0
...
0000 0
...
9988 0
...
9887 0
...
9502 0
...
8555
6
1
...
0000 0
...
9996 0
...
9964 0
...
9819 0
...
0000 1
...
0000 1
...
9999 0
...
9993 0
...
9961
8
1
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...
0000 0
...
0000 0
...
0000 0
...
0000
7
0
...
0009 0
...
0002 0
...
0000 0
...
0000 0
...
0120 0
...
0043 0
...
0010 0
...
0001 0
...
0000
9
0
...
0468 0
...
0196 0
...
0048 0
...
0002 0
...
2487 0
...
1595 0
...
0809 0
...
0231 0
...
0000
11
0
...
5814 0
...
4596 0
...
3062 0
...
1136 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=13
x
p=
0
...
02
p=
0
...
04
p=
0
...
06
p=
0
...
08
p=
0
...
8775 0
...
6730 0
...
5133 0
...
3893 0
...
2935
1
0
...
9730 0
...
9068 0
...
8186 0
...
7206 0
...
9997 0
...
9938 0
...
9755 0
...
9422 0
...
8946
3
1
...
9999 0
...
9986 0
...
9940 0
...
9837 0
...
0000 1
...
0000 0
...
9997 0
...
9987 0
...
9959
5
1
...
0000 1
...
0000 1
...
9999 0
...
9997 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
9999
7
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
9
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
11
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
13
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
15
p=
0
...
25
p=
0
...
35
p=
0
...
45
p=
0
...
2542 0
...
0550 0
...
0097 0
...
0013 0
...
0001
1
0
...
3983 0
...
1267 0
...
0296 0
...
0049 0
...
8661 0
...
5017 0
...
2025 0
...
0579 0
...
0112
3
0
...
8820 0
...
5843 0
...
2783 0
...
0929 0
...
9935 0
...
9009 0
...
6543 0
...
3530 0
...
1334
5
0
...
9925 0
...
9198 0
...
7159 0
...
4268 0
...
9999 0
...
9930 0
...
9376 0
...
7712 0
...
5000
7
1
...
9998 0
...
9944 0
...
9538 0
...
8212 0
...
0000 1
...
9998 0
...
9960 0
...
9679 0
...
8666
9
1
...
0000 1
...
9999 0
...
9975 0
...
9797 0
...
0000 1
...
0000 1
...
9999 0
...
9987 0
...
9888
11
1
...
0000 1
...
0000 1
...
0000 0
...
9995 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
9999
13
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
1
0
...
0001 0
...
0000 0
...
0000 0
...
0000 0
...
0041 0
...
0003 0
...
0000 0
...
0000 0
...
0000
3
0
...
0078 0
...
0007 0
...
0000 0
...
0000 0
...
0698 0
...
0126 0
...
0010 0
...
0000 0
...
0000
214
5
0
...
0977 0
...
0182 0
...
0012 0
...
0000 0
...
3563 0
...
1295 0
...
0243 0
...
0013 0
...
0001
7
0
...
4256 0
...
1654 0
...
0300 0
...
0009 0
...
7721 0
...
4995 0
...
2060 0
...
0342 0
...
0041
9
0
...
8314 0
...
5794 0
...
2527 0
...
0342 0
...
9731 0
...
8868 0
...
6674 0
...
3080 0
...
1054
11
0
...
9874 0
...
9363 0
...
7664 0
...
3787 0
...
9996 0
...
9963 0
...
9762 0
...
8791 0
...
7065
13
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
2
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
4
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
6
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0003 0
...
0001 0
...
0000 0
...
0000 0
...
0000
8
0
...
0013 0
...
0003 0
...
0000 0
...
0000 0
...
0163 0
...
0060 0
...
0014 0
...
0001 0
...
0000
10
0
...
0578 0
...
0245 0
...
0062 0
...
0003 0
...
2794 0
...
1814 0
...
0932 0
...
0270 0
...
0000
12
0
...
6107 0
...
4867 0
...
3270 0
...
1225 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
215
n=14
x
p=
0
...
03
p=
0
...
05
p=
0
...
07
p=
0
...
09
0
0
...
7536 0
...
5647 0
...
4205 0
...
3112 0
...
9916 0
...
9355 0
...
8470 0
...
7436 0
...
6368
2
0
...
9975 0
...
9833 0
...
9522 0
...
9042 0
...
0000 0
...
9994 0
...
9958 0
...
9864 0
...
9685
4
1
...
0000 1
...
9998 0
...
9990 0
...
9965 0
...
0000 1
...
0000 1
...
0000 0
...
9998 0
...
9992
6
1
...
0000 1
...
0000 1
...
0000 1
...
0000 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
8
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
10
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
12
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
14
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
01
p=
0
...
15
p=
0
...
25
p=
0
...
35
p=
0
...
45
p=
0
...
2288 0
...
0440 0
...
0068 0
...
0008 0
...
0001
1
0
...
3567 0
...
1010 0
...
0205 0
...
0029 0
...
8416 0
...
4481 0
...
1608 0
...
0398 0
...
0065
3
0
...
8535 0
...
5213 0
...
2205 0
...
0632 0
...
9908 0
...
8702 0
...
5842 0
...
2793 0
...
0898
5
0
...
9885 0
...
8883 0
...
6405 0
...
3373 0
...
9998 0
...
9884 0
...
9067 0
...
6925 0
...
3953
7
1
...
9997 0
...
9897 0
...
9247 0
...
7414 0
...
0000 1
...
9996 0
...
9917 0
...
9417 0
...
7880
9
1
...
0000 1
...
9997 0
...
9940 0
...
9574 0
...
0000 1
...
0000 1
...
9998 0
...
9961 0
...
9713
11
1
...
0000 1
...
0000 1
...
9999 0
...
9978 0
...
0000 1
...
0000 1
...
0000 1
...
9999 0
...
9991
13
1
...
0000 1
...
0000 1
...
0000 1
...
0000 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
60
p=
0
...
70
p=
0
...
80
p=
0
...
90
p=
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
1
0
...
0001 0
...
0000 0
...
0000 0
...
0000 0
...
0022 0
...
0001 0
...
0000 0
...
0000 0
...
0000
3
0
...
0039 0
...
0002 0
...
0000 0
...
0000 0
...
0426 0
...
0060 0
...
0003 0
...
0000 0
...
0000
5
0
...
0583 0
...
0083 0
...
0004 0
...
0000 0
...
2586 0
...
0753 0
...
0103 0
...
0003 0
...
0000
7
0
...
3075 0
...
0933 0
...
0116 0
...
0002 0
...
6627 0
...
3595 0
...
1117 0
...
0115 0
...
0008
9
0
...
7207 0
...
4158 0
...
1298 0
...
0092 0
...
9368 0
...
7795 0
...
4787 0
...
1465 0
...
0315
11
0
...
9602 0
...
8392 0
...
5519 0
...
1584 0
...
9971 0
...
9795 0
...
8990 0
...
6433 0
...
3632
13
0
...
9992 0
...
9932 0
...
9560 0
...
7712 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
217
x
p=
0
...
93
p=
0
...
95
p=
0
...
97
p=
0
...
99
p=
1
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
1
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
3
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
5
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
7
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0004 0
...
0001 0
...
0000 0
...
0000 0
...
0000
9
0
...
0020 0
...
0004 0
...
0000 0
...
0000 0
...
0214 0
...
0080 0
...
0019 0
...
0001 0
...
0000
11
0
...
0698 0
...
0301 0
...
0077 0
...
0003 0
...
3100 0
...
2037 0
...
1059 0
...
0310 0
...
0000
13
0
...
6380 0
...
5123 0
...
3472 0
...
1313 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
n=15
x
p=
0
...
02
p=
0
...
04
p=
0
...
06
p=
0
...
08
p=
0
...
8601 0
...
6333 0
...
4633 0
...
3367 0
...
2430
1
0
...
9647 0
...
8809 0
...
7738 0
...
6597 0
...
9996 0
...
9906 0
...
9638 0
...
9171 0
...
8531
3
1
...
9998 0
...
9976 0
...
9896 0
...
9727 0
...
0000 1
...
9999 0
...
9994 0
...
9972 0
...
9918
218
5
1
...
0000 1
...
0000 0
...
9999 0
...
9993 0
...
0000 1
...
0000 1
...
0000 1
...
0000 0
...
9998
7
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
9
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
11
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
13
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
15
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
10
p=
0
...
20
p=
0
...
30
p=
0
...
40
p=
0
...
50
0
0
...
0874 0
...
0134 0
...
0016 0
...
0001 0
...
5490 0
...
1671 0
...
0353 0
...
0052 0
...
0005
2
0
...
6042 0
...
2361 0
...
0617 0
...
0107 0
...
9444 0
...
6482 0
...
2969 0
...
0905 0
...
0176
4
0
...
9383 0
...
6865 0
...
3519 0
...
1204 0
...
9978 0
...
9389 0
...
7216 0
...
4032 0
...
1509
6
0
...
9964 0
...
9434 0
...
7548 0
...
4522 0
...
0000 0
...
9958 0
...
9500 0
...
7869 0
...
5000
8
1
...
9999 0
...
9958 0
...
9578 0
...
8182 0
...
0000 1
...
9999 0
...
9963 0
...
9662 0
...
8491
10
1
...
0000 1
...
9999 0
...
9972 0
...
9745 0
...
0000 1
...
0000 1
...
9999 0
...
9981 0
...
9824
12
1
...
0000 1
...
0000 1
...
9999 0
...
9989 0
...
0000 1
...
0000 1
...
0000 1
...
0000 0
...
9995
14
1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
65
p=
0
...
75
p=
0
...
85
p=
0
...
91
0
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0001 0
...
0000 0
...
0000 0
...
0000 0
...
0000
2
0
...
0003 0
...
0000 0
...
0000 0
...
0000 0
...
0063 0
...
0005 0
...
0000 0
...
0000 0
...
0000
4
0
...
0093 0
...
0007 0
...
0000 0
...
0000 0
...
0769 0
...
0124 0
...
0008 0
...
0000 0
...
0000
6
0
...
0950 0
...
0152 0
...
0008 0
...
0000 0
...
3465 0
...
1132 0
...
0173 0
...
0006 0
...
0000
8
0
...
3902 0
...
1311 0
...
0181 0
...
0003 0
...
7392 0
...
4357 0
...
1484 0
...
0168 0
...
0013
10
0
...
7827 0
...
4845 0
...
1642 0
...
0127 0
...
9576 0
...
8273 0
...
5387 0
...
1773 0
...
0399
12
0
...
9729 0
...
8732 0
...
6020 0
...
1841 0
...
9983 0
...
9858 0
...
9198 0
...
6814 0
...
3965
14
0
...
9995 0
...
9953 0
...
9648 0
...
7941 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
x
p=
0
...
92
p=
0
...
94
p=
0
...
96
p=
0
...
98
p=
0
...
00
0
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
2
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
4
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
6
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0000
8
0
...
0000 0
...
0000 0
...
0000 0
...
0000 0
...
0007 0
...
0001 0
...
0000 0
...
0000 0
...
0000
10
0
...
0028 0
...
0006 0
...
0001 0
...
0000 0
...
0273 0
...
0104 0
...
0024 0
...
0002 0
...
0000
12
0
...
0829 0
...
0362 0
...
0094 0
...
0004 0
...
3403 0
...
2262 0
...
1191 0
...
0353 0
...
0000
14
0
...
6633 0
...
5367 0
...
3667 0
...
1399 0
...
0000 1
...
0000 1
...
0000 1
...
0000 1
...
0000
Poisson Distribution Tables
221
Title: business statistics
Description: The above apploded files are my class notes for last semister. Best in the class of of business and economics students
Description: The above apploded files are my class notes for last semister. Best in the class of of business and economics students